PRAISE FOR AMIR D. ACZEL'S

FERMAT'S LAST THEOREM

"This is a captivating volume. . . . The brilliant backdoor method

used by Mr. Wiles as he reached his solution, along with the debt he

owed to many other contemporary mathematicians, is graspable in

Mr. Aczel's lucid prose. Equally important is the sense of awe that

Mr. Aczel imparts for the hidden, mystical harmonies of numbers,

and for that sense of awe alone, his slender volume is well worth the

effort."

—The New York Times

"For more than three centuries, Fermat's Last Theorem was the

most famous unsolved problem in mathematics, here's the story of

how it was solved. . . . An excellent short history of mathematics,

viewed through the lens of one of its great problems— and

achievements."

—Kirkus Reviews

"This exciting recreation of a landmark discovery reveals the great

extent to which modern mathematics is a collaborative

enterprise. . . . While avoiding technical details, Aczel maps the

strange, beautiful byways of modern mathematical thought in ways

the layperson can grasp."

—Publishers Weekly

"Briefly chronicles the history of the famous problem of the title,

which was recently solved by a mathematician named Andrew Wiles

after he had devoted seven years to the task. . . . Aczel does a

superb job of creating in the nonmathematical reader the illusion of

comprehension."

—The New Yorker

Pierre de Fermat (1601-1665).

FERMAT'S LAST THEOREMUnlocking the Secret ojan Ancient

Mathematical Problem

By Amir D. Aczel

D e l t a

A Delta Book

Published by Dell

Publishing a

division of

Bantam Doubleday Dell Publishing Group, Inc.

1540 Broadway

New York, New York 10036

Copyright © 1996 by Amir D Aczel

All rights reserved. No part of this book may be reproduced or transmitted in any

form or by any means, electronic or mechanical, including photocopying,

recording, or by any information storage and retrieval system, without the written

permission of the Publisher, except where permitted by law. For information

address: Four Walls Eight Windows, New York, NY, 10011.

The trademark Delta® is registered in the U.S. Patent and Trademark Office and

in other countries.

ISBN: 0-385-31946-0

Reprinted by arrangement with Four Walls Eight Windows

Manufactured in the United States of America Published

simultaneously in Canada

October 1997

10 9 8 7 6 5 4 3 2 1

BVG

Preface

In June 1993, my old friend Tom Schulte was visiting me in

Boston from California. We were sitting at a sunny sidewalk cafe on

Newbury Street, tall, icy drinks in front of us. Tom had just gotten

divorced, and he was ruminative. He half turned toward me. "By the

way," he said, "Fermat's Last Theorem has just been proved." This

must be a new joke, I thought, as Tom's attention was back on the

sidewalk. Twenty years earlier, Tom and I were roommates, both of

us undergraduate students in mathematics at the University of

California at Berkeley. Fermat's Last Theorem was something we

often talked about. We also discussed functions, and sets, and

number fields, and topology. None of the math students slept much

at night, since our assignments were so difficult. That's what

distinguished us from students in most other areas. Sometimes

we'd have math nightmares . . . trying to prove some theorem or

another before it was due in the morning. But Fermat's Last

Theorem? No one ever believed it would be proven in our lifetime.

The theorem was so difficult, and so many people had tried to

prove it for over three hundred years. We were well aware that

entire branches of mathematics had been developed as the result

of attempts to prove the theorem. But the attempts failed, one by

one. Fermat's Last Theorem had come to symbolize the unat-

tainable. I once even used the theorem's perceived impossibility to

my advantage. It was a few years later, also at Berkeley, when I

had already graduated in math and was getting my Master's in

operations research. An arrogant graduate student

in mathematics, unaware of my own background in math, offered

me his help when we met at the International House where we both

lived. "I'm in pure mathematics," he said. "If you ever have a math

problem you can't solve, feel free to ask me." He was about to

leave, when I said "Urn, yes. There is something you can help me

with . . . " He turned back, "Why, sure, let me see what it is." I

pulled open a napkin—we were at the dining room. I slowly wrote

on it:

xn + yn = zn has no whole number solution when n is greater

than 2.

"I've been trying to prove this one since last night," I said,

handing him the napkin. I could see the blood draining from his

face. "Fermat's Last Theorem," he grunted. "Yes," I said, "You're in

pure math. Could you help me?" I never saw that person from up

close again.

"I am serious," Tom said, finishing his drink. "Andrew Wiles. He

proved Fermat's Last Theorem in Cambridge last month. Remember

that name. You'll hear it a lot." That night, Tom was in the air, flying

back to California. In the next months, I realized Tom didn't play a

joke on me, and I followed the sequence of events where Wiles was

at first applauded, then the hole in his proof was found, then he

withdrew for a year, and finally reemerged with a corrected proof.

But following the continuing saga, I learned that Tom was wrong. It

wasn't Andrew Wiles's name I should have paid attention to, or at

least not his alone. I, and the world, should have recognized that

the proof of Fermat's Last Theorem was far from the work of one

mathematician. While Wiles got much of the praise,

the accolades belong to others as much: Ken Ribet, Barry Mazur,

Goro Shimura, Yutaka Taniyama, Gerhard Frey and others. This

book tells the entire story, including what happened behind the

scenes and out of the view of the media's cameras and floodlights.

It is also a story of deception, intrigue, and betrayal.

IN

"Perhaps I could best describe my experience of doing

mathematics in terms oj entering a dark mansion. You go into the

first room and it's dark, completely dark. You stumble around,

bumping into the furniture. Gradually, you learn where each piece

of furniture is. And finally, after six months or so, you find the light

switch and tum it on. Suddenly, it's all illuminated and you can see

exactly where you were. Then you enter the next dark room . . . "

This is how Professor Andrew Wiles described bis seven-year

Quest for the mathematicians' Holy Grail.

Just before dawn on June 23, 1993, Professor John Conway

approached the darkened mathematics building on the Princeton

University campus. He unlocked the front door and quickly walked

up to his office. For weeks preceding his colleague Andrew Wiles'

departure for England, persistent but unspecific rumors had been

circulating in the world's mathematical community. Conway was

expecting something important to happen. Exactly what it was, he

had no idea. He turned on his computer and sat down to stare at

the screen. At 5:53 AM, a terse e-mail message flashed from

across the Atlantic: "Wiles proves FL.T."

Cambridge, England, June 1993

Late in June of 1993, Professor Andrew Wiles flew to England. He

was returning to Cambridge University, where he had been a

graduate student twenty years earlier. Wiles' former doctoral thesis

adviser at Cambridge, Professor John Coates, was organizing a

conference on Iwasawa Theory—the particular area within number

theory in which Andrew Wiles did his dissertation and about which

he knew a great deal. Coates had asked his former student if he

would mind giving a short, one-hour talk at the conference on a

topic of his choice. To his great surprise and that of the other

conference organizers, the shy Wiles—previously reluctant to speak

in public—responded by asking if he could be given three hours of

presentation.

The 40-year-old Wiles looked the typical mathematician when he

arrived in Cambridge: white dress shirt with sleeves rolled up

carelessly, thick horn-rimmed glasses, unruly strands

of thinning light hair. Born in Cambridge, his return was a very

special kind of homecoming—it was the realization of a childhood

dream. In pursuit of this dream, Andrew Wiles had spent the last

seven years of his life a virtual prisoner in his own attic. But he

hoped that soon the sacrifice, the years of struggle and the long

hours of solitude would end. Soon he might be able to spend more

time with his wife and daughters, of whom he had seen so little for

seven years. He had often failed to show up for lunch with his

family, missed afternoon tea, barely made it to dinner. But now the

accolades would be his alone.

The Sir Isaac Newton Institute for Mathematical Sciences at

Cambridge had only recently opened by the time Professor Wiles

arrived to deliver his three hour-long lectures. The Institute is

spacious, set in scenic surroundings at some distance from the

University of Cambridge. Wide areas outside the lecture halls are

furnished with plush, comfortable chairs, designed to help facilitate

the informal exchange of ideas among scholars and scientists, and

to promote learning and knowledge.

Although he knew most of the other mathematicians who came

to the specialized conference from around the world, Wiles kept to

himself. When colleagues became curious about the length of his

scheduled presentation, Wiles would only say they should come to

his lectures and find out for themselves. Such secretiveness was

unusual, even for a mathematician. While they often work alone

trying to prove theorems and are generally not known to be the

world's most gregarious people, mathematicians usually share

research results with each other. Mathematical results are freely

circulated by their authors in the form of research preprints. These

preprints bring their authors outside

comments that help them improve the papers before they are

published. But Wiles didn't hand out preprints and didn't discuss his

work. The title of Wiles' talks was "Modular Forms, Elliptic Curves,

and Galois Representations," but the name gave no hint where the

lectures would lead, and even experts in his field could not guess.

The rumors intensified as time went on.

On the first day, Wiles rewarded the 20 or so mathematicians

who came to his lecture with a powerful and unexpected

mathematical result—and there were still two more lectures to go.

What was coming? It became clear to everyone that Wiles' lectures

were the place to be, and the suspense grew as expectant

mathematicians flocked to the lectures.

On the second day, Wiles' presentation intensified. He had

brought with him over 200 pages of formulas and derivations,

original thoughts stated as new theorems with their lengthy,

abstract proofs. The room was now filled to capacity. Everyone

listened intently. Where would it lead? Wiles gave no hint. He

dispassionately continued writing on the blackboard and when he

was done for the day, he quickly disappeared.

The next day, Wednesday, June 23, 1993, was his last talk.

As he neared the lecture hall, Wiles found it necessary to push his

way in. People stood outside blocking the entrance and the room

was overflowing. Many carried cameras. As Wiles again wrote

seemingly endless formulas and theorems on the board, the

tension increased. "There was only one possible climax, only one

possible end to Wiles' presentation," Professor Ken Ribet of the

University of California at Berkeley later told me. Wiles was

finishing the last few lines of his proof of an enigmatic and

complicated conjecture in mathematics, the

Shimura-Taniyama Conjecture. Then suddenly he added one final

line, a restatement of a centuries-old equation, one which Ken Ribet

had proved seven years earlier would be a consequence of the

conjecture. "And this proves Fermat's Last Theorem," he said,

almost offhandedly. "I think I'll stop here."

There was a moment of stunned silence in the room. Then the

audience (erupted in spontaneous applause. Cameras flashed as

everyone stood up to congratulate a beaming Wiles. Within

minutes, electronic mail flashed and faxes rolled out of machines

around the world. The most celebrated mathematical problem of all

time appeared to have been solved.

"What was so unexpected was that the next day we were

deluged by the world press," recalled Professor John Coates, who

organized the conference without having the slightest idea that it

would become the launching ground for one of the greatest

mathematical achievements. Headlines in the world's newspapers

hailed the unexpected breakthrough. "At Last, Shout of 'Eureka!' In

Age-Old Math Mystery" announced the front page of the New York

Times on June 24, 1993. The Washington Post called Wiles in a

major article "The Math Dragon-Slayer," and news stories

everywhere described the person who apparently solved the most

persistent problem in all of mathematics, one that had defied

resolution for over 350 years. Overnight, the quiet and very

private Andrew Wiles became a household name.

Pierre de Fermat

Pierre de Fermat was a seventeenth-century French jurist who was

also an amateur mathematician. But while he was technically

an "amateur" since he had a day job as a jurist, the leading histo-

rian of mathematics E. T. Bell, writing in the early part of the

twentieth century, aptly called Fermat the "Prince of Amateurs."

Bell believed Fermat to have achieved more important mathe-

matical results than most "professional" mathematicians of his day.

Bell argued that Fermat was the most prolific mathematician of the

seventeenth century, a century that witnessed the work of some of

the greatest mathematical brains of all time.1

One of Fermat's most stunning achievements was to develop the

main ideas of calculus, which he did thirteen years before the birth

of Sir Isaac Newton. Newton and his contemporary Gottfried

Wilhelm von Leibniz are jointly credited in the popular tradition with

having conceived the mathematical theory of motion, acceleration,

forces, orbits, and other applied mathematical concepts of

continuous change we call calculus.

Fermat was fascinated with the mathematical works of ancient

Greece. Possibly he was led to his conception of calculus ideas by

the work of the classical Greek mathematicians Archimedes and

Eudoxus, who lived in the third and fourth centuries B.C.,

respectively. Fermat studied the works of the ancients—which were

translated into Latin in his day—in every spare moment. He had a

full-time job as an important jurist, but his hobby—his passion—was

to try to generalize the work of the ancients and to find new beauty

in their long-buried discoveries. "I have found a great number of

exceedingly beautiful theorems," he once said. These theorems he

would jot down in the margins of the translated copies of ancient

books he possessed.

Fermat was the son of a leather merchant, Dominique Fer-

Pierre de Fermat's "Last Theorem" as reproduced in an edition of Diophantus'

Arithmetica published by Fermat's son Samuel. The original copy of Diophantus with

Fermat's handwritten note has never been found.

mat, who was Second Consul in the town of Beaumont-de-Lomagne,

and of Claire de Long, the daughter of a family of parliamentary

judges. The young Fermat was born in August, 1601 (baptized

August 20 in Beaumont-de-Lomagne), and was raised by his

parents to be a magistrate. He went to school in Toulouse, and was

installed in the same city as Commissioner of Requests at the age

of thirty. He married Louise Long, his mother's cousin, that same

year, 1631. Pierre and Louise had three sons and two daughters.

One of their sons, Clement Samuel, became his father's scientific

executor and published his father's works after his death. In fact, it

is the book containing Fermat's work, published by his son, that has

come down to us and from which we know his famous Last

Theorem. Clement Samuel de Fermat recognized the importance of

the theorem scribbled in the margin and added it to the translation

of the ancient work he republished.

Fermat's life is often described as quiet, stable, and uneventful.

He did his work with dignity and honesty, and in 1648 was

promoted to the important position of the King's Councillor-ship in

the local Parliament of Toulouse, a title he held for seventeen years

until his death in 1665. Considering the great work Fermat did for

the Crown, a lifetime of devoted, able, and conscientious service,

many historians are puzzled that he had the time and the mental

energy to do first-rate mathematics—and volumes of it. One French

expert suggested that Fermat's official work was actually an asset

to his mathematical studies, since French parliamentary councilors

were expected to minimize their unofficial contacts in order to

avoid the temptations of bribery and other corruption. Since Fermat

certainly required a

diversion from his hard work, and since he had to limit his social

life, mathematics probably offered a much-needed break. And the

ideas of calculus were far from Fermat's only achievement. Fermat

brought us number theory. An important element in number theory

is the concept of a prime number.

Prime Numbers

The numbers two and three are prime numbers. The number four is

not prime because it is the product of two and two: 2 x 2 = 4.

The number five is prime. The number six is not prime since, like

four, it is the product of two numbers: 2 x 3 = 6. Seven is

prime, eight is not (2 x 2 x 2 = 8), nine is not (3 x 3=9), and

ten is not (2 x 5=10). But eleven again is a prime number since

there are no integers (other than eleven itself and one), which can

be multiplied together to give us 1 1 . And we can continue this

way: 12 is not prime, 13 is, 14 is not, 15 is not, 16 is not, 17

is prime, and so on. There is no apparent structure here, such as

every fourth number is not a prime, or even any more complicated

pattern. The concept has mystified human beings since early

antiquity. Prime numbers are the essential elements in number

theory, and the lack of easily-seen structure tends to make number

theory seem un-unified as a field, and its problems isolated, difficult

to solve, and without clear implications to other fields of

mathematics. In the words of Barry Mazur: "Number theory

produces, without effort, innumerable problems which have a

sweet, innocent air about them, tempting flowers,- and yet . . .

number theory swarms with bugs, waiting to bite the tempted

flower-lovers who, once bitten, are inspired to excesses of effort!"2

A Famous Note on the Margin

Fermat was smitten by the charm of numbers. In them he found

beauty and meaning. He came up with a number of theorems in

number theory, one of which was that every number of the form

22An+l (two raised to the power two raised to the power n, plus one)

is a prime number. Later, it was discovered that the theorem was

false when a number of this form was found not to be a prime.

Among Fermat's cherished Latin translations of ancient texts was

a book called the Arithmetka, written by the Greek mathematician

Diophantus, who lived in Alexandria in the third century A.D. Around

1637, Fermat wrote in Latin in the margin of his Diophantus, next

to a problem on breaking down a squared number into two squares:

On the other hand, it is impossible to separate a cube into two

cubes, or a biquadrate into two biquadrates, or generally any

power except a square into two powers with the same exponent.

I have discovered a truly marvelous proof of this, which,

however, the margin is not large enough to contain.

This mysterious statement kept generations of mathematicians

busy trying to supply the "truly marvelous proof" Fermat claimed to

have possessed. The statement itself, that while a square of a

whole number could be broken down into two other squares of

whole numbers (for example, five squared, which is twenty-five,

equals the sum of four squared (sixteen) and three squared [nine]),

but that the same cannot be done with cubes or any higher powers,

looked deceptively simple.

All of Fermat's other theorems were either proved or disproved by

the early 1800s. This seemingly simple statement remained

unsettled, and therefore was given the name "Fermat's Last The-

orem." Was it indeed true? Even in our own century, computers

were stymied in attempts to verify that the theorem was true.

Computers could verify the theorem for very large numbers, but

they couldn't help for all numbers. The theorem could be tried on

billions of numbers, and there still would be infinitely many—and

infinitely many exponents—to check. To establish Fermat's Last

Theorem, a mathematical proof was required. Awards were offered

in the 1800s by the French and German scientific academies to

anyone who would come up with a proof, and every year thousands

of mathematicians and amateurs, along with cranks, sent "proofs"

to mathematical journals and judging committees—always coming

up empty-handed.

July-August, 1993—A Fatal Flaw is Discovered Mathematicians were

cautiously optimistic when Wiles stepped down from the podium

that Wednesday in June. Finally, the 350 -year-old mystery seemed

to have been solved. Wiles' lengthy proof, using complicated

mathematical notions and theories which were not known during

the time of Fermat or indeed until the twentieth century, needed to

be validated by independent experts. The proof was sent to a

number of leading mathematicians. Perhaps seven years of working

alone in the seclusion of his attic had finally paid off for Wiles. But

the optimism was short-lived. Within weeks, a hole was discovered

in Wiles' logic. He tried to patch it, but the gap would simply not go

away. Princeton mathematician Peter Sarnak, a

close friend of Andrew Wiles, watched him agonize every day over

the proof he had told the entire world he possessed only two

months earlier in Cambridge. "It's as if Andrew was trying to lay an

over-sized carpet on the floor of a room," Sarnak explained. "He'd

pull it out, and the carpet would fit perfectly in one side of the

room, but across the room it would be up against the wall, so he

would go there and pull it down . . . and then it would pop up in

another place. Whether or not the carpet had the right size for the

room was not something he was able to determine." Wiles withdrew

into his attic. The reporters from the New York Times and the rest

of the media left him to his lonely task. As time went by without a

proof, mathematicians and the public in general began to wonder

whether Fermat's theorem was at all true. The marvelous proof

Professor Wiles convinced the world he possessed became no more

real than Fermat's own "truly marvelous proof which the margin is

unfortunately too small to contain."

Between the Tigris and Euphrates Rivers, Circa 2 0 0 0 B.C. The story

of Fermat's Last Theorem is much, much older than Fermat himself.

It's even older than Diophantus, whose work Fermat was trying to

generalize. The origins of this simple-looking yet profound theorem

are as old as human civilization itself. They are rooted in the Bronze

Age culture that developed in the Fertile Crescent between the

Tigris and Euphrates rivers of ancient Babylon (an area within

today's Iraq). And while Fermat's Last Theorem is an abstract

statement with no applications in science, engineering,

mathematics—not even in number theory, its own niche within

mathematics—the

roots of this theorem are grounded in the everyday life of the

people of Mesopotamia of 2000 B.C. .

The era from 2000 B.C. to 600 B.C. in the Mesopotamian

valley is considered the Babylonian Era. This time saw remarkable

cultural developments, including writing, the use of the wheel, and

metal works. A system of canals was used for irrigating large tracts

of land between the two rivers. As civilization flourished in the

fertile valley of Babylon, the ancient people who inhabited these

planes learned to trade and to build prosperous cities such as

Babylon and Llr (where Abraham was born). Even earlier, by the

end of the fourth millennium B.C., a primitive form of writing had

already developed in both the Mesopotamian and the Nile river

valleys. In Mesopotamia, clay was abundant and wedge-shaped

marks were impressed with a stylus on soft clay tablets. These

tablets were then baked in ovens or left to harden in the sun. This

form of writing is called cuneiform, a word derived from the Latin

word cuneus, meaning wedge. The cuneiform constitutes the first

known writing the world has ever seen. Commerce and

construction in Babylon and in ancient Egypt brought the need for

accurate measurements. The early scientists of these Bronze Age

societies learned to estimate the ratio between the circumference

and the diameter of a circle, which gave them a number close to

what we call today pi. The people who built the giant Ziggurat, the

biblical Tower of Babel, and the Hanging Gardens, one of the Seven

Wonder of the Ancient World, needed a way to compute areas and

volumes.

Wealth Is a Squared Quantity

A sophisticated number system was developed using base sixty and

Babylonian engineers and builders were able to compute the

quantities required in their everyday professional lives. Squares of

numbers appear naturally in life, although it doesn't seem so at first

glance. Squares of numbers can be viewed as representing wealth.

A farmer's prosperity is dependent on the amount of crops he is

able to produce. These crops, in turn, depend on the area that is

available to the farmer. The area is a product of the length and the

width of the field, and this is where squares come in. A field that

has length and width equal to a has area equal to a-squared. In this

sense, therefore, wealth is a squared quantity.

The Babylonians wanted to know when such squares of whole

numbers could be partitioned into other squares of whole numbers.

A farmer who owned one field of twenty-five square units of land

could swap it for two square fields: one measuring sixteen squared

units and the other nine squared units. So a field five units by five

units was equivalent to two fields, one four by four and the other

three by three. This was important information for the solution of a

practical problem. Today we would write this relationship in the

form of an equation: 52 = 3 2 + 4 2 . And triples of such integers,

here 3, 4, and 5, whose squares satisfy this relation, are called

Pythagorean triples—even though they were known to the

Babylonians over one thousand years before the time of the famous

Greek mathematician, Pythagoras, after whom they are named. We

know all this from an unusual clay tablet dated to around 1900

B.C.

F E R M A T ' S L A S T T H E O R E M

"Plimpton 322"

The Babylonians were obsessed with tables. And the abundance of

clay and the cuneiform writing technology they possessed allowed

them to create many of them. Because of the clay tablets'

durability, many of them survive today. From one location alone,

the site of ancient Nippur, over 50,000 tablets were recovered

and are now in the collections at the museums of Yale, Columbia,

and the University of Pennsylvania, among others. Many of these

tablets are in the basements of the muse-' urns, gathering dust,

lying there unread and undeciphered.

One tablet that was deciphered is remarkable. This tablet, in the

museum of Columbia University, is called Plimpton 322. All it

contains are 15 triples of numbers. Each one of the triples has the

property that the first number is a square and is the sum of the

other two, each being itself a squared number—the table contains

fifteen Pythagorean triples.3 The numbers 25 = 16 + 9, given

earlier, form a Pythagorean triple. Another Pythagorean triple on

Plimpton 322 is 169 = 144 + 25 (13 2 = 122 + 5 2). Not

all scholars agree on the reason for the ancient Babylonians'

interest in these numbers. One theory is that the interest was solely

for practical purposes, and the fact that they used a number system

with base sixty and therefore preferred integers to fractions

supports this need to solve practical problems with nice, whole

square numbers. But other experts think that an inherent interest in

numbers themselves may also have been a motivator for the

Babylonians' interest in square numbers. It seems that, whatever

the motive, Plimpton 322 may have served as a tool for teaching

students to solve problems where the numbers are perfect squares.

The Babylonians' approach was not to develop a general theory

for solving such problems, but rather to provide tables listing triples

of numbers and—apparently—to teach pupils how to read and use

these tables.

An Ancient Society of Number-Worshippers Sworn to Secrecy

Pythagoras was born on the Greek island of Samos around 580 B.C.

He travelled extensively throughout the ancient world and visited

Babylon, Egypt, and possibly even India. In his travels, especially in

Babylon, Pythagoras came in contact with mathematicians and likely

became aware of their studies of numbers now named after him—

the Pythagorean triples,

which Babylonian scientists and mathematicians had known about

for over 1500 years. Pythagoras came in contact with the

builders of magnificent works of art and architecture, and the

mathematical aspects of these wonders could not escape him.

Pythagoras was also exposed in his travels to religious and

philosophical ideas of the East.

When Pythagoras returned to Greece he left the island of Samos

and moved to Crotona, on the Italian "boot." It is interesting to note

that Pythagoras certainly saw most of the Seven Wonders of the

Ancient World. One of these wonders, the Temple of Hera, is right

where Pythagoras was born on Samos. Today, the ruins of the

magnificent temple—only one standing column remains from

among hundreds—are a short walk away from the modern town of

Pythagorion, named in honor of the island's illustrious son. Just

across the strait and a few miles to the north, in modern-day

Turkey, lie another of the Wonders in the remains of ancient

Ephesus. The Colossus of Rhodes is nearby, to the south of Samos;

the Pyramids and the Sphynx are in Egypt and Pythagoras saw

those,- and in Babylon he must have seen the Hanging Gardens.

The Italian boot, including Crotona where Pythagoras settled, as

well as much of the rest of southern Italy, were at that time part

of the Greek world—Magna Graecia. This "greater Greece" included

settlements al l over the eastern Mediterranean, including

Alexandria in Egypt with its large ethnic Greek population—

descendants of which remained there through the early 1900s.

Not far from Crotona were caves for oracles just like the Oracle of

Delphi, who was believed to foretell fortunes and futures of people

and nations.

A M I R D . A C Z E L

"Number Is

Everything"

In the barren, stark surroundings of the tip of Italy, Pythagoras

founded a secret society dedicated to the study of numbers. The

society, whose members became known collectively as the

Pythagoreans, is believed to have developed a substantial body of

mathematical knowledge—all in complete secrecy. The

Pythagoreans are believed to have followed a philosophy sum-

marized by their motto that "number is everything." They wor-

shipped numbers and believed them to have magical properties. An

object of interest to them was a "perfect" number. One of the

definitions of a perfect number—a concept that continued to be

pursued in the Middle Ages, and appears in mystical systems such

as the Jewish Kabbalah—is a number that is the sum of its

multiplicative factors. The best and simplest example of a perfect

number is the number six. Six is the product of three and two and

one. These are the multiplicative factors of this number, and we

have: 6 = 3 x 2 x 1 . But note also that if you add the same

factors you will again get the number six: 6 = 3 + 2 + 1 . In

that sense, six is "perfect." Another perfect number is 28, since

the numbers that can divide 28 (without remainder) are 1, 2, 4,

7, and 14, and we note that also: 1 + 2 + 4 + 7 + 14 =

28.

The Pythagoreans followed an ascetic lifestyle, and were strict

vegetarians. But they did not eat beans, thinking they resembled

testicles. Their preoccupations with number were very much in the

spirit of a religion, and their strict vegetarianism originated in

religious beliefs. While no documents survive dating to the time of

Pythagoras, there is a large body of later literature about the

master and his followers, and Pythagoras himself is considered one

of the greatest mathematicians of

antiquity. To him is attributed the discovery of the Pythagorean

Theorem concerning the squares of the sides of a right triangle,

which has strong bearing on Pythagorean triples and, ultimately, on

Fermat's Last Theorem two thousand years later.

The Square oj the Hypotenuse Is Equal to the

Sum oj the Squares oj the Other Two Sides. . .

The theorem itself originated in Babylon, since the Babylonians

clearly understood "Pythagorean" triples. The Pythagoreans,

however, are credited with setting the problem in geometric

terms, and thus generalizing them away from strictly the natural

numbers (positive integers without zero). The Pythagorean

Theorem says that the square of the hypotenuse of a right-triangle

is equal to the sum of the squares of the two remaining sides of the

triangle, as shown above.

When the hypotenuse is an integer (such as 5, whose square is

25), the general Pythagorean solution in terms of the sum of two

squares will be the integers four (whose square is sixteen) and

three (whose square is nine). So the Pythagorean Theorem, when

applied to integers (whole numbers such as 1, 2, 3, . . . )

gives us the Pythagorean triples which were known a millennium

earlier in Babylon.

Incidentally, the Pythagoreans also knew that squared numbers

are sums of sequences of odd numbers. For example, 4 = 1 +

3,- 9 = 1 + 3 + 5,- 16= 1 + 3 + 5 + 7, and so on. This

property they represented by a visual array of numbers in a square

pattern. When the odd number of dots along two adjacent sides is

added to the previous square, a new square is formed:

Whole Numbers, Fractions, and What Else?

But the Pythagoreans knew a lot more than whole numbers and

fractions (numbers such as 1/2, 1/3, 5/8, 147/1769, etc.),

which were known in antiquity both in Babylon and in Egypt. The

Pythagoreans are the ones who discovered the irrational numbers—

that is, numbers that cannot be written as fractions but have to be

written as unending, non-repeating decimals. One such example is

the number pi (3.141592654 . . . ), the ratio of the

circumference of a circle to its diameter. The number pi is

unending,- it would take forever to write it down completely since it

has infinitely many (that is, neverending) distinct digits. To write it

down, we simply say: pi ( K ) . Or, we can write it to any finite

number of decimals, such as 3.14, 3.1415, or so. Computers

were used in our century to compute and write down pi to a million

or more digits, but this is rarely necessary. Pi was known to within

various approximations to the Babylonians and Egyptians of the

second millennium B.C. They took it as roughly three, and it arose

naturally as a consequence of the discovery of the wheel. Pi also

arose in various measurements of a pyramid. Pi is even alluded to

in the Old Testament: in Kings I, 7:23, we read about a circular

wall being constructed. From the given number of units for the

circumference and the diameter, we can conclude that the ancient

Israelites took pi to be roughly three.

The Pythagoreans discovered that the square root of two was an

irrational number. From an application of the Pythagorean Theorem

to a right triangle with two sides both equal to one unit, the

Pythagoreans obtained that the hypotenuse was a strange number:

the square root of two. They could tell that this

number was not an integer, nor even a fraction, a ratio of two

integers. This was a number with an unending decimal represen-

tation which did not repeat itself. As is the case with pi, to write

down the exact number that is the square root of two

(1.414213562 . . . ) would take forever, since there are infi-

nitely many digits, forming a unique sequence (rather than a

repeating one such as

1.857142857142857142857142857 . . . , etc., which

one could describe without having to actually write down every

digit). Any number with a repeating decimal representation (here,

the sequence 857142 repeats itself over and over again in the

decimal part of this number), is a rational number, that is, a

number that can also be written in the form alb, that is, the ratio of

two integers. In this example, the two integers are 13 and 7. The

ratio 13/7 is equal to 1.857142857142857142857142 . .

. , the pattern 857142 repeating itself forever.

The discovery of the irrationality of the square root of two

surprised and shocked these diligent number admirers. They swore

never to tell anyone outside their society. But word got out. And

legend has it that Pythagoras himself killed by drowning the

member who divulged to the world the secret of the existence of

the strange, irrational numbers.

The numbers on the number line are of two distinct kinds:

rational and irrational. Looked at together, they fill the entire line

with no holes. The numbers are very, very close (inflnites-imally

close) to each other. The rational numbers are said to be

everywhere dense within the real numbers. Any neighborhood, any

tiny interval, around a rational number contains infinitely many of

these irrational numbers. And vice versa,

around every irrational number there are infinitely many rational ones.

Both sets, the rational and the irrational numbers, are infinite. But the

irrationals are so numerous that there are more of them than there are

rational numbers. Their order of infinity is higher. This fact was shown

in the 1800s by the mathematician Georg Cantor (1845-1918). At

the time, few people believed Cantor. His arch-enemy Leopold

Kronecker (1823-1891) taunted and ridiculed Cantor for his

theories about how many rational and irrational numbers- there are.

Kronecker is known for his statement, "God made the integers, and all

the rest is the work of man," meaning that he did not even believe that

the irrational numbers, such as the square root of two, existed! (This,

over two millennia after the Pythagoreans.) His antagonism is blamed

for having prevented Cantor from obtaining a professorship at the

prestigious University of Berlin, and ultimately for Cantor's frequent

nervous breakdowns and his ending up at an asylum for the mentally

infirm. Today, all mathematicians know that Cantor was right and that

there are infinitely many more irrational numbers than rational ones,

even though both sets are infinite. But did the ancient Greeks know

that much?4

F E R M A T ' S L A S T T H E O R E M The

Pythagorean Legacy

An important aspect of Pythagorean life, with its dietary rules and

number worship and secret meetings and rituals, was the pursuit of

philosophical and mathematical studies as a moral basis. It is

believed that Pythagoras himself coined the words philosophy: love

of wisdom,- and mathematics-, that which is learned. Pythagoras

transformed the science of mathematics into a liberal form of

education.

Pythagoras died around 500 B.C. and left no written records of

his work. His center at Crotona was destroyed when a rival political

group, the Sybaritics, surprised the members and murdered most

of them. The rest dispersed about the Greek world around the

Mediterranean, carrying with them their philosophy and number

mysticism. Among those who learned the philosophy of

mathematics from these refugees was Philolaos of Tarentum, who

studied at the new center the Pythagoreans established in that city.

Philolaos is the first Greek philosopher to have written down the

history and theories of the Pythagorean order. It is from the book

written by Philolaos that Plato learned of the Pythagorean

philosophy of number, cosmology, and mysticism, about which he

later wrote himself. The special symbol of the Pythagorean order

was the five-pointed star embedded in a pentagon. The diagonals

which form the five-pointed star intersect in such a way that they

form another, smaller pentagon, in a reversed direction. If the

diagonals inside this smaller pentagon are drawn, they form yet

another pentagon, and so on ad infinitum. This pentagon and five-

pointed star made up of its diagonals have some fascinating

properties, which the Pythagoreans believed

were mystical. A diagonal point divides a diagonal into two unequal

parts. The ratio of the entire diagonal to the larger segment is

exactly the same as the ratio of the larger segment to the smaller

one. This same ratio exists in all smaller and smaller diagonals. This

ratio is called the Golden Section. It is an irrational number equal to

1.618 . . . If you divide 1 by this number, you get exactly the

decimal part without the 1 . That is, you get 0.618. . . . As we

will see later, the Golden Section appears in natural phenomena as

well as proportions that the human eye perceives as beautiful. It

appears as the limit of the ratio of the famous Fibonacci numbers

we will soon encounter.

You can find the Golden Section by an interesting sequence of

operations on a calculator. Do 1 + 1 =, then press 1 /x, then +1

= , then 1 /x, then +1 = , then l/x and keep going. The number on

your display should become 1.618.. . and 0.618.. . inter-

changeably, once you have done the repetitive set of operations

enough times. This is the Golden Section. It is equal to the square

root of five, minus 1, divided by 2. This is the way it is obtained

geometrically from the Pythagorean Pentagon. Since this ratio

never becomes a ratio of two integers, hence never a rational

number, it proves that the square root of five is also an irrational

number. We will see more of the Golden Section later.

The Pythagoreans discovered that harmony in music corre-

sponded to simple ratios of numbers. According to Aristotle, the

Pythagoreans believed that all of heaven was musical scale and

numbers. Musical harmony and geometrical designs are what

brought the Pythagoreans to their belief that "All is number." The

Pythagoreans thought that the basic ratios in music involved only

the numbers 1, 2, 3, and 4, whose sum is 10. And 10, in

turn, is the base of our number system. The Pythagoreans

represented the number 10 as a triangle, which they called

tetraktys-.5

The Pythagoreans considered the tetraktys holy and even swore

oaths by it. Incidentally, according to Aristotle as well as Ovid and

other classical writers, the number ten was chosen as the base for

the number system because people have ten fingers. Recall that

the Babylonians, on the other hand, used a number system based

on sixty. There are some remnants of other number systems even

today. The French word for eighty (quatre-vingt, meaning "four

twenties") is a relic of an archaic number system based on twenty.

The Ropes, The Nile, and the Birth oj Geometry Much of what we

know about ancient Greek mathematics comes from the Elements

of Euclid of Alexandria, who lived around 300 B.C. It is believed

that the first two volumes of the Elements are all on the work of

Pythagoras and his secret society. The mathematics of the ancient

Greeks was done for its beauty and concerned abstract geometrical

figures. The Greeks developed an entire theory of geometry and it

is this theory, mostly unchanged, that is taught in schools today. In

fact, the Elements, or what remains of it today, is considered the

greatest textbook of all time.

Herodotus, the great Greek historian of antiquity, believed that

geometry was developed in ancient Egypt of 3,000 B.C., long

before the Greeks of Alexandria and elsewhere. He tells how the

overflow of the Nile would destroy boundaries between fields in the

river's fertile delta, and how this necessitated complicated

surveying techniques. For the purpose of this work, the surveyors

had to develop geometrical concepts and ideas. In his Histories,

Herodotus writes:

If the river carried away any portion of a man's lot, the king sent persons to

examine and determine by measurement the exact extent of the loss. From

this practice, I think, geometry first came to be known in Egypt, whence it

passed into Greece.6

Geometry is the study of shapes and figures made of circles and

straight lines and arcs and triangles and their intersections forming

various angles. It stands to reason that such science would be

essential for good surveying work. Egyptian geometers were indeed

called "rope stretchers," since ropes were used for outlining

straight lines necessary both in building temples and pyramids and

in realigning boundaries between fields. But it is possible that the

origins of geometry are even more ancient. Neolithic finds show

examples of congruence and symmetry of design, and these may

have been the precursors of Egyptian geometry, inherited centuries

later by the ancient Greeks. The same concerns that the

Babylonians had with areas of fields, leading to their need to

understand square numbers and their relations, may have been

shared by the ancient Egyptians, who were faced with the same

agrarian quandaries as well as construction problems with their

own pyramids. It is possible, therefore, that the ancient Egyptians

also had a knowledge of Pythagorean triples. What the Greeks did

with geometry, however, was to establish it as a pure mathematical

endeavor. They postulated and proved theorems.

What is a Theorem?

The Greeks brought us the concept of a theorem. A theorem is a

mathematical statement whose proof is given. The proof of a

theorem is a rigorous justification of the veracity of the theorem in

such a way that it cannot be disputed by anyone who follows the

rules of logic, and who accepts a set of axioms put forth as the

basis for the logic system. Euclid's axioms include the definition of

a point, a line, and the statement that two parallel lines never

meet. Following axioms and logical progressions, such as if A

implies B and B implies C then A implies C, the ancient Greeks were

able to prove many beautiful theorems about the geometry of

triangles and circles and squares and octagons and hexagons and

pentagons.

"Eureka! Eureka!"

The great Greek mathematicians Eudoxus (fifth century B.C.) and

Archimedes (third century B.C.) extended such work on geometrical

figures to the finding of areas using infinitesimal (meaning infinitely

small) quantities. Eudoxus of Cnidus (408-355 B.C.) was a friend

and student of Plato. He was too poor to live in the Academy in

Athens, and so he lived in the cheaper harbor town of Piraeus, from

which he commuted daily to Plato's Academy. While Plato himself

was not a mathematician, he encouraged mathematical work,

especially that of gifted students such as Eudoxus. Eudoxus

traveled to Egypt and there, as well as in Greece, learned much

geometry. He invented a "method of exhaustion," which he used to

find areas of geometric figures by using infinitesimal quantities.

Eudoxus would, for example, approximate the area of a circle by

the sum of the areas of many small rectangles—whose areas are

easy to calculate as the base times the height. This is essentially

the method used today in integral calculus, and the mod

em limit arguments are not different from Eudoxus' "exhaustion"

method.

But the most brilliant mathematician of antiquity was

undoubtedly Archimedes (287-212 B.C.), who lived in the city of

Syracuse on the island of Sicily. Archimedes was the son of the

astronomer Pheidias, and was related to Hieron II, the king of

Syracuse. Like Eudoxus, Archimedes developed methods for finding

areas and volumes, which were the forerunners of calculus. His

work anticipated both integral calculus and differential calculus

(there are two parts to calculus—Archimedes understood both of

them). But while he was mostly interested in pure mathematics:

numbers, geometry, areas of geometric figures, and so on, he is

also known for his achievements in the applications of

mathematics. A well-known story is the one about Archimedes'

discovery of what we call today the first law of hydrostatics: the law

that a submerged body loses from its weight the weight of the

liquid it displaces. There was at that time a dishonest goldsmith in

Syracuse, and King Hieron asked

his mathematician friend to find a way to prove this. Archimedes

started by studying the loss of weight of submerged bodies, using

his own body in the experiments. He took a bath and made some

measurements. When he discovered the law, he jumped out of the

bath and ran naked through the streets of Syracuse shouting

"Eureka, eureka!" ("I found it, I found it!").

Archimedes is also credited with the discovery of archimedes'

screw, a device to raise water by turning a hand-operated crank. It

is still in use by farmers around the world.

When the Roman General Marcellus attacked Syracuse in 214-

212 B.C., Hieron again asked for the help of his illustrious relative.

As the Roman fleet was approaching, Archimedes devised great

catapults based on his study of levers, and the Syracusans were

able to defend themselves well. But Marcellus regrouped his forces

and some time later attacked from the rear and was able to take

Syracuse by surprise. This time Archimedes was not even aware of

the attack and was sitting quietly on the ground above the city

drawing geometrical figures in the sand. A Roman soldier

approached and stepped on the figures. Archimedes jumped,

exclaiming: "Don't disturb my circles!" at which the soldier drew his

sword and killed the 75-year-old mathematician. In his will,

Archimedes had apparently requested that his gravestone bear a

carving of the particular geometric figure he admired—a sphere

inside a cylinder. The neglected grave was covered and the site

lost, but the Roman orator Cicero found it many years later and

restored it, and then the sands of time covered it again. In 1963,

ground was broken for a new hotel near Syracuse, and there

workers rediscovered Archimedes' tomb.

Archimedes' favorite theorem had to do with the sphere inside

the cylinder and he wrote the theorem in a book called The Method.

As with most ancient texts, it was assumed lost. In 1906, the

Danish scholar J. L. Heiberg heard that in Constantinople there was

a faded parchment manuscript with writings of a mathematical

nature. He traveled to Constantinople and found the manuscript,

consisting of 185 leaves of parchment. Scientific studies proved it

was a tenth-century copy of Archimedes' book, over which Eastern

Orthodox prayers had been added in the thirteenth century.

Alexandria, Greek Egypt, Circa A.D. 250

Around A.D. 250 a mathematician by the name of Diophantus

lived in Alexandria. All that we know about the life of Diophantus is

what is given in the following problem in a collection called the

Palatine Anthology, written roughly a century after Diophantus'

death.7

Here you see the tomb containing the remains of Diophantus, it is

remarkable: artfully it tells the measures of his life. The sixth part of his life

God granted him for his youth. After a twelfth more his cheeks were

bearded. After an additional seventh he kindled the light of marriage, and in

the fifth year he accepted a son. Alas, a dear but unfortunate child, half of

his father he was when chill Fate took him. He consoled his grief in the

remaining four years of his life. By this devise of numbers, tell us the extent

of his life.

(If you solve the implied equation, you will find that the answer

is 84.)

It is not certain when Diophantus lived. We can date his period

based only on two interesting facts. First, he quotes in his writings

Hypsicles, who we know lived around 150 B.C. And second,

Diophantus himself is quoted by Theon of Alexandria. Theon's time

is dated well by the solar eclipse which occurred on June 16, A.D.

364. So Diophantus certainly lived before A.D. 364 but after

150 B.C. Scholars, somewhat arbitrarily, place him at about A.D.

250.

Diophantus wrote the Arithmetica, which developed algebraic

concepts and gave rise to a certain type of equation. These are the

Diophantine equations, used in mathematics today. He wrote

fifteen volumes, only six of which came down to us. The rest were

lost in the fire that destroyed the great library of Alexandria, the

most monumental collection of books in antiquity. The volumes that

survived were among the last Greek texts to be translated. The first

known Latin translation was published in 1575. But the copy

Fermat had was the one translated by Claude Bachet in 1621 . It

was Diophantus' Problem 8 in Volume II, asking for a way of

dividing a given square number into the sum of two squares—the

Pythagorean problem whose solution was known to the Babylonians

two thousand years earlier—that inspired Fermat to write his

famous Last Theorem in the margin. The mathematical

achievements of Diophantus and his contemporaries were the final

glory of the ancient Greeks.

Arabian Nights

While Europe was busy fighting little feudal wars of the vassals of

one king or prince against another, surviving the Great Plague, and

going on costly and often deadly expeditions

called the Crusades, the Arabs ruled a flourishing empire from the

Middle East to the Iberian Peninsula. Among their great

achievements in medicine, astronomy, and the arts, the Arabs

developed algebra. In A.D. 632, the prophet Mohammed

established an Islamic state centered at Mecca, which remains the

religious center of Islam. Shortly afterwards, his forces attacked the

Byzantine Empire, an offensive which continued after Mohammed's

death in Medina that same year. Within a few years, Damascus,

Jerusalem, and much of Mesopotamia fell to the forces of Islam, and

by A.D. 641 so did Alexandria—the mathematical center of the

world. By A.D. 750, these wars as well as the ones among the

Moslems themselves subsided and the Arabs of Morocco and the

west were reconciled with the eastern Arabs centered in Baghdad.

Baghdad became a center of mathematics. The Arabs absorbed

mathematical ideas as well as discoveries in astronomy and other

sciences from the inhabitants of the areas they overcame. Scholars

from Iran, Syria, and Alexandria were called to Baghdad. During the

reign of the caliph Al Mamun in the early 800s, the Arabian Nights

was written and many Greek works, including Euclid's Elements,

were translated into Arabic. The caliph established a House of

Wisdom in Baghdad, and one of its members was Mohammed Ibn

Musa Al-Khowarizmi. Like Euclid, Al-Khowarizmi was to become

world-renowned. Borrowing Hindu ideas and symbols for numerals,

as well as Mesopotamian concepts and Euclid's geometrical

thought, Al-Khowarizmi wrote books on arithmetic and algebra. The

word "algorithm" is derived from Al-Khowarizmi. And the word

"algebra" is derived from the first words in the title of Al-

Khowarizmi's most well-known book: Al Jabr Wa'l Mucjabalah. It was

from this book that Europe was later to learn the branch of

mathematics called algebra. While algebraic ideas are in the root of

Diophantus' Aritbmetica, the Al Jabr is more closely related to the

algebra of today. The book is concerned with straightforward

solutions of equations of first and second degree. In Arabic, the

name of the book means "restoration by transposing terms from

one side of an equation to the other"— the way first-order

equations are solved today.

Algebra and geometry are related, as are all branches of

mathematics. One field that links these two together is algebraic

geometry, developed in our century. It is the linkage of branches of

mathematics, and areas that lie within different branches and

connect them, that would pave the way to Wiles' work on Fermat's

problem centuries later.

The Medieval Merchant and the Golden Section

The Arabs were interested in a problem that was very closely

related to the Diophantine question of finding Pythagorean triples.

The problem was to find Pythagorean triples giving an area of a

right-triangle that is also an integer. Hundreds of years later, this

problem found itself as the basis for the book Liber Quadratorum,

written in 1225 by Leonardo of Pisa (1180-1250). Leonardo

was better known as Fibonacci (which means "son of Bonaccio").

Fibonacci was an international merchant born in Pisa. He also lived

in North Africa and Constantinople, and he traveled extensively

throughout his life and visited Provence, Sicily, Syria, Egypt, and

many other areas in the Mediterranean. His travels and his relations

with the elite of Mediterranean soci

ety of the time brought him into contact with Arab mathematical

ideas, as well as Greek and Roman culture. When the emperor

Frederick II came to Pisa, Fibonacci was introduced to the emperor's

court and became a member of the imperial entourage.

In addition to Liber Quadratorum, Fibonacci is known for another

book he wrote at that time, Liber Abaci. A problem about

Pythagorean triangles in Fibonacci's book also appears in a

Byzantine manuscript of the eleventh century now in the Old Palace

library in Istanbul. It could be a coincidence,- on the other hand,

Fibonacci might have seen that same book in Constantinople during

his travels.

Fibonacci is best known for the sequence of numbers named

after him, the Fibonacci Numbers. These numbers originate in the

following problem in the Liber Abaci-.

How many pairs of rabbits will be produced in a year, beginning with a

single pair, if in every month each pair bears a new pair which becomes

productive from the second month on?

The Fibonacci sequence, which is derived from this problem, is

one where each term after the first is obtained by adding together

the two numbers that precede it. The sequence is= 1, 1 ,2, 3, 5,

8, 13, 21, 34, 55, 89, 144, . . .

This sequence (which is taken to continue beyond the 12

months of the problem) has unexpectedly significant properties.

Amazingly, the ratio of two successive numbers in the sequence

tends to the Golden Section. The ratios are: 1/1, 1 /2, 2/3, 3/5,

5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144, etc.

Note that these numbers get closer and closer to (sqr5 - l)/2.

This is the Golden Section. It can also be obtained using a cal-

culator by repeating the operation l/l + l/l + 1/. . . . as described

earlier. Recall that the reciprocal (l/x) of the Golden Section gives

the same number, less 1 . The Fibonacci sequence appears

everywhere in nature. Leaves on a branch grow at distances from

one another that correspond to the Fibonacci sequence. The

Fibonacci numbers occur in flowers. In most flowers, the number of

petals is one of: 3, 5, 8, 13, 2 1 , 34, 55, or 89. Lilies have

three petals, buttercups five, delphiniums often eight, marigolds

thirteen, asters twenty-one, daisies usually thirty-four or fifty-five or

eighty-nine.

The Fibonacci numbers appear in sunflowers too. The little

florets that become seeds in the head of the sunflower are

arranged in two sets of spirals: one winding in a clockwise direction

and the other counter-clockwise. The number of spirals in the

clockwise orientation is often thirty-four and the counter-clockwise

fifty-five. Sometimes the numbers are fifty-five and eighty-nine, and

sometimes even eighty-nine and a hundred and forty-four. All are

consecutive Fibonacci numbers (whose ratio approaches the golden

section). Ian Stewart argues in Nature's Numbers that when spirals

are developed, the angles between them are 137.5 degrees,

which is 360 degrees multiplied by one minus the golden ratio,

and they also give rise to two successive Fibonacci numbers for the

number of clockwise and counter-clockwise spirals, as shown

below.8

If a rectangle is drawn with sides in the Golden Section ratio to

each other, then the rectangle can be divided into a square and

another rectangle. This second rectangle is similar to the large one

in that it, too, has ratio of sides equal to the Golden Section. The

smaller rectangle can now be divided into a square and a remaining

rectangle, also in the Golden ratio . . . and so on. A spiral through

successive vertices of the sequence of rectangles that can be

drawn is one that appears often in shells, in the arrangement of

sunflower florets as mentioned, and in the arrangement of leaves

on a branch.

The rectangle has appealing proportions. The Golden Section

appears not only in nature but also in art as the classic ideal of

beauty. There is something divine about the sequence, and in fact

the Fibonacci Society, which is active today, is headed by a priest

and centered at St. Mary's College in California. The Society is

dedicated to the pursuit of examples of the Golden Section and

Fibonacci Numbers in nature, art, and architecture, with the belief

that the ratio is a gift of God to the world. As the ideal of beauty,

the Golden Section appears in such places as the Athenian

Parthenon. The ratio of the height of the Parthenon to its length is

the Golden Section.

The great Pyramid at Giza, built many hundreds of years before

the Greek Parthenon, has ratio of height of a face to half the side of

the base also in the Golden Section. The Egyptian Rhind Papyrus

refers to a "sacred ratio." Ancient statues as well as renaissance

paintings display the proportions equal to the Golden Section, the

Divine Ratio.

The Golden Section has been searched for as the ideal of beauty

beyond flowers or architecture. In a letter to the Fibonacci Society

some years ago, a member described how someone looking to find

the Golden Section asked several couples to perform an

experiment. The husband was asked to measure the height of his

wife's navel and divide it by his wife's height. The writer claimed

that in all couples the ratio was close to 0.618.

The Cossists

Mathematics entered Medieval Europe through Fibonacci's works

and from Spain, then part of the Arab world, with the work of Al

Khowarizmi. The main idea of algebra in those days was to solve

equations for an unknown quantity. Today, we call the unknown

quantity "x," and try to solve an equation for whatever value "x"

may have. An example of the simplest equation is: x - 5 = 0.

Here, we use simple math operations to find the value of "x." If we

add 5 to both sides of the equation we get on the left side x - 5 +

5, and on the right side we get 0 + 5. So the left side is "x" and

the right side is 5. That is, x = 5. The Arabs of Al Khowarizmi's

day called the unknown quantity "thing." The word "thing" in Arabic

is shai. And so they solved equations for the unknown shai, as done

above with "x."

When these ideas were imported to Europe, the Arabic shai was

translated into Latin. In Latin, "thing" is res, and in Italian it is cosa.

Since the first European algebraists were Italian, the word cosa was

attached to them. Since they were concerned with solving

equations for an unknown cosa, they became known as the

Cossists.

As in Babylon three and a half millennia earlier, mathematics in

the Middle Ages and the early Renaissance was mainly used as an

aid in commerce. Mercantile society of that time was increasingly

concerned with problems of trade, exchange rates, profits, costs,

and these could sometimes be cast as mathematical problems

requiring the solution of some equation. The cossists were people

such as Luca Pacioli (1445-1514), Geronimo Cardano (1501-

1576), Niccolo Tartaglia (1500-1557), and others who

competed with each other as problem-solvers in the service of

merchants and traders. These mathematicians used the solution of

more abstract problems as a form of advertising. Since they had to

compete for clients, they would also spend time and effort solving

these more difficult problems, such as cubic equations (equations

where the unknown quantity "cosa," or our "x," is in the third

power, x3), so that they could publish the results and become ever

more sought-after for the solution of applied problems.

In the early 1500s, Tartaglia found a way to solve cubic

equations and kept the method secret, so he could retain an edge

over his competitors in the lucrative problem-solving market. After

Tartaglia won a problem-solving competition against another

mathematician, Cardano pressed him to reveal his secret of how he

could solve these cubic equations.

Tartaglia revealed his method, on the condition that Cardano keep

it a secret from the rest of the world. When Cardano later learned

the same methods from another cossist, Scippi-one del Ferro

(1456-1526), he immediately assumed that Tartaglia got his

system from this person, and felt free to reveal the secret. Cardano

then published the methodology for solving cubic equations in his

1545 book Ars Magna. Tartaglia felt betrayed and became furious

with Cardano. In his last years, he spent much of his time vilifying

his former friend, and he succeeded in diminishing Cardano's

reputation.

The cossists were considered mathematicians of a lower level

than the ancient Greeks. Their preoccupation with applied problems

in pursuit of financial success, and their unconstructive fights

among themselves, kept them from looking for beauty in

mathematics and the pursuit of knowledge for its own sake. They

did not develop an abstract, general theory of mathematics. For

that, one needed to go back to the ancient Greeks. That is exactly

what happened a century later.

Renaissance Quest for Ancient Knowledge

Thirteen hundred years had passed since Diophantus. The medieval

world gave way to the Renaissance and the beginning of the

modern age. Out of the darkness of the Middle Ages, Europe awoke

with a thirst for knowledge. Many people turned their interest to

classical works of the ancients. Whatever ancient books existed

were translated into Latin—the language of the educated—in this

revival of the search for knowledge and enlightenment. Claude

Bachet, a French nobleman, was a translator with a great interest in

mathematics. He

obtained a copy of the Greek Arithmetica of Diophantus, translated

it and published it as Diophanti Alexandrini Arithmeticorum Libri Sex

in Paris in 1621. It was a copy of this book that found its way to

Fermat.

Fermat's theorem says that there are no possible Pythagorean

triples for anything beyond squares. There are no triples of

numbers, two adding up to the third, where the three numbers are

perfect cubes of integers, or fourth-powers of integers, fifth, sixth, or

any other powers. How could Fermat come up with such a theorem?

Squares, Cubes, and Higher Dimensions

A theorem is a statement with a proof. Fermat claimed to have had

a "marvelous proof," but without seeing and validating the proof no

one could call his statement a theorem. A statement may be very

deep, very meaningful and important, but without the proof that it is

indeed true, it must be called a conjecture, or sometimes a

hypothesis. Once a conjecture is proven it then can be called a

theorem, or a lemma if it is a preliminary proven statement which

then leads to a more profound theorem. Proven results that follow a

theorem are called corollaries. And Fermat had a number of such

statements. One such statement was that the number 22 A | 1+1 was

always a prime number. This conjecture was not only not proven,

hence not a theorem, it was actually proven to be wrong. This was

done by the great Swiss mathematician Leonhard Euler (1707-

1783) in the following century. So there was no reason to believe

that the "last theorem" was true. It could be true, or it could be

false. To prove that Fermat's Last Theorem was false all some-

one would have to do is to find a triple of integers, a , b , and c, and

a power n , greater than 2, that satisfy the relation a" + b" = c".

No one has ever found such a set of integers. (However, assuming

that a solution exists was to be a key element in attempts to prove

the theorem later.) And by the 1990s it was shown that no such

integers exist for any n less than four million. But that did not

mean that such numbers could not be found some day. The

theorem had to be proven for all integers and all possible powers.

Fermat himself was able to prove his Last Theorem for «=4. He

used an ingenious method he called the method of "infinite

descent," to prove that no integers a , b , and c exist satisfying a 4

+ b4 = c4. He also recognized that i f a solution existed for any

power n , then it would also exist for any multiple of n. One would

therefore only have to consider prime numbers (greater than 2) as

the exponents, that is , numbers that cannot be divided by any

number other than 1 or themselves in integers. The first few prime

numbers are 1, 2, 3, 5, 7, 1 1 , 1 3 , 17. .. None of these

numbers can be divided by any number other than 1 or themselves

and an integer result. An example of a number that is not a prime

is 6, since 6 divided by 3 gives 2—an integer. Fermat was also

able to prove his theorem for n = 3. Leon-hard Euler proved the

case n = 3 and n = 4 independently of Fermat, and Peter G. L.

Dirichlet in 1828 was able to prove the case n = 5. The same

case was proved by Adrien-Marie Legendre in 1830. Gabriel

Lame, and Henri Lebesgue who corrected him in 1840, were able

to establish the case n=7. Thus, two hundred years after Fermat

wrote his famous note in the margin of his Diophantus, his

theorem was only proven

correct for the exponents 3, 4, 5, 6, and 7. It would be a long

way to infinity, which is what one had to do to prove the theorem

for any exponent n. Clearly, what was needed was a general proof

that would work for all exponents, however large they might be.

Mathematicians were all looking for the elusive general proof, but

unfortunately what they were finding were proofs for particular

exponents only.

The Algorist

An algorist is a person who devises computing systems, or

algorithms. One such person was the prolific Swiss mathematician

Leonhard Euler, who was said to be able to calculate as naturally as

other people breathed. But Euler was much more than a walking

calculator. He was the most productive Swiss scientist of all times,

and a mathematician who wrote so many volumes of work that the

Swiss government established a special fund to collect all of his

works. He is said to have produced mathematical papers between

two calls for dinner at his large household.

Leonhard Euler was born in Basel on April 15, 1707. The fol-

lowing year, the family moved to the village of Riechen, where the

father became the Calvinist pastor. When young Leonhard went to

school, his father encouraged him to pursue the study of theology

so that he would eventually take his place as the village pastor. But

Euler showed much promise in mathematics and was tutored by

Johannes Bernoulli, a well-known Swiss mathematician of the day.

Daniel and Nicolaus Bernoulli, two younger members of the large

Bernoulli family of mathematicians, became his good friends. These

two convinced Leon-

hard's parents to allow him to pursue mathematics, since he would

become a great mathematician. Leonhard, however, continued with

the theology in addition to mathematics, and religious feelings and

customs would be a part of his entire life.

Mathematicial and scientific research in Europe in those days did

not take place primarily at universities, as it does today. The

universities were more devoted to teaching and did not allow much

time for other activities. Research in the eighteenth century was

primarily done at royal academies. There, the monarch would

support the leading scientists of the day in their pursuit of

knowledge. Some of the knowledge was applied, and would help

the government improve the standing of the entire nation. Other

research was more "pure," that is, research for its own sake—for

the advancement of human knowledge. The royals supported such

research generously and the scientists working at the academies

were able to lead comfortable lives.

When he finished his studies of mathematics, as well as theology

and Hebrew, at the University of Basel, Euler applied for a

professorship. Despite the great achievements he had already

made, he was turned down. In the meantime, his two friends Daniel

and Nicolaus were appointed as research mathematicians at the

royal academy in St. Petersburg, Russia. The two stayed in touch

with Leonhard and promised that, somehow, they would get him

there too. One day, the two Bernoullis wrote an urgent letter to

Euler telling him that there was an opening in the medical section

of the St. Petersburg academy. Euler immediately put himself to

work studying physiology and medicine at Basel. Medicine was not

something that interested him, but he was desperate to have a job

and hoped that

this way he could join his two friends who had such excellent

positions doing nothing but their own research in Russia.

Euler found mathematics in whatever he studied, medicine

included. Studying ear physiology led him to a mathematical

analysis of the propagation of waves. At any rate, soon an invitation

came from St. Petersburg, and in 1727 he joined his two friends.

However, on the death of Peter's wife Catherine, there was chaos at

the academy since she had been the great supporter of research. In

the confusion, Leonhard Euler slipped out of the medical section

and somehow got his name listed with the mathematical section,

where he would rightfully belong. For six years he kept his head low

to prevent the detection of his move, and he avoided all social

interactions lest the deception be discovered. All through this

period he worked continuously, producing volumes of top-rated

mathematical work. In 1733 he was promoted to the leading

mathematical position at the academy. Apparently Euler was a

person who could work anywhere, and as his family was growing,

he would often do his mathematics while holding a baby in one

arm.

When Anna Ivanova, Peter the Great's niece, became empress of

Russia, a period of terror began and Euler again hid himself in his

work for ten years. During this period he was working on a difficult

problem in astronomy for which a prize was offered in Paris. A

number of mathematicians requested several months' leave from

the academy to work on the problem. Euler solved it in three days.

But the concentrated exertion took its toll and he became blind in

his right eye.

Euler moved to Germany to be at the royal academy there, but

did not get along with the Germans, who enjoyed long

philosophical discussions not to his taste. Catherine the Great of

Russia invited Euler back to the St. Petersburg academy and he was

more than happy to come back. At that time, the philosopher Denis

Diderot, an atheist, was visiting Catherine's court. The empress

asked Euler to argue with Diderot about the existence of God.

Diderot, in the meantime, was told that the famous mathematician

had a proof of God's existence. Euler approached Diderot and said

gravely: "Sir, a + bin = x, hence God exists,- reply!" Diderot, who

knew nothing about mathematics, gave up and immediately

returned to France.

During his second stay in Russia, Euler went blind in his second

eye. He continued, however, to do mathematics with the help of his

sons, who did the writing for him. Blindness increased his mental

ability to do complicated calculations in his head. Euler continued to

do mathematics for seventeen years and died while playing with his

grandson in 1783. Much of the mathematical notation we use

today is due to Euler. This includes the use of the letter i for the

basic imaginary number, the square root of - 1 . Euler loved one

mathematical formula, which he considered the most beautiful and

put it above the gates of the Academy. The formula is:

e* + 1 = 0

This formula has 1 and 0, basic to our number system,- it has the

three mathematical operations: addition, multiplication, and

exponentiation,- and it has the two natural numbers pi and e,- and

it has i, the basis for the imaginary numbers. It is also visually

appealing.

The Seven Bridges oj Konigsberg

Euler was such an incredible visionary in mathematics that his

pioneering work on imaginary numbers (and what today is called

complex analysis), was not his only innovation. He did pioneering

work in a field which, in our century, would become indispensable

in the work of mathematicians—and in attempts to solve the

Fermat mystery. The field is topology, a visual theory of spatial

configurations that can remain unchanged when transformed by

continuous functions. It is the study of shapes and forms, some with

intricate, unexpected geometry, which is extended to four, five, or

higher dimensions beyond our normal three-dimensional world. We

will visit this fascinating area again when we get to the modern

approach to Fermat's problem, since topology—much as it seems

unrelated to the Fermat equation—has great importance for

understanding it.

Predating the development of topology, Euler's contribution to

the field is the famous problem of the Seven Bridges of Konigsberg.

This is the puzzle that started the whole interest in topology. In

Euler's time, seven bridges crossed the Pregel River in Konigsberg.

These are shown in the diagram below.

Euler asked whether or not it was possible to cross all seven

bridges without passing twice on any bridge. It is impossible. Other

problems, which were studied in modern times and were posed

because of interest in the seven bridges problem, are the various

map-coloring problems. A cartographer draws a map of the world.

In this map, each country is colored differently, to distinguish it

from its immediate neighbors. Any two countries or states that are

separated completely from each other may be colored in the same

exact color. The question is, what is the minimum number of colors

required so that no two states that touch each other are in the

same color? Of course, this is a general problem, not bounded by

how the map of the world looks today. The question is really, given

all possible configurations of maps on a plane, what is the minimum

number of colors that can be used? Given boundaries between

states in the former Yugoslavia or in the Middle East, with very

unusual lines between political entities, this general problem

becomes relevant in applications.

Mathematically, this is a topological problem. In October 1852,

Francis Guthrie was coloring a map of England. He wondered what

would be the minimum number of colors to be used for the

counties. It occurred to him that the number should be four. In

1879 a proof was given that the number was indeed four, but

later the proof was found to be false. Almost a century later, in

1976, two mathematicians, Haken and Appel, proved what had

become known as the Four Color Map problem. To this day,

however, their proof is considered controversial since it made use

of computer work, rather than pure mathematical logic.

Gauss, The Great German Genius

An alleged error in Euler's proof for n = 3 (that is, for cubes) was

corrected by Carl Friedrich Gauss (1777-1855). While most of

the renowned mathematicians of this time were French, Gauss,

undoubtedly the greatest mathematician of the time—and arguably

of all times—was unmistakably German. In fact, he never left

Germany, even for a visit. Gauss was the grandson of a very poor

peasant, and the son of a laborer in Brunswick. His father was harsh

with him, but his mother protected and encouraged her son. Young

Carl was also taken care of by his uncle Friedrich, the brother of

Gauss' mother Dorothea. This uncle was wealthier than Carl's par-

ents and made a reputation for himself in the field of weaving.

When Carl was three years old, he once watched his uncle add up

his accounts in a ledger. "Uncle Friedrich," he interrupted, "this

calculation is wrong." The uncle was shocked. From that day on, the

uncle did everything possible to contribute to the education and

care of the young genius. Although Gauss showed incredible

promise in school, his behavior sometimes left something to be

desired. One day, the teacher punished young Gauss by telling him

to stay in the classroom until he added up all the numbers from 1

to 100 while everyone else went to play outside. Two minutes

later, the 10-year-old Gauss was outside playing with the rest of

the class. The teacher came out furious. "Carl Friedrich!" the

teacher called, "do you want a more severe punishment? I told you

to stay inside until you have finished adding up all the numbers!"

"But I have," he said, "here is the answer." Gauss handed the

teacher a piece of paper with the right answer,

5,050, written on it. Apparently, Gauss figured out that he could

write two rows of 101 numbers:

0 1 2 3..... 97 98 99 100

100 99 98 97..... 3 2 1 0

He noted that the sum of each column was 100, so there was

nothing long to add up. Since there were 101 columns, the sum of

all the numbers was 101 x 100=10,100. Now, either of the two

rows had the sum he needed—all the numbers from 1 to 100.

Since he needed only one of the two rows, the answer was half of

10,100, or 5,050. Very simple, he thought. The teacher,

however, learned a lesson and never again assigned the young

Gauss a math problem as punishment.

When he was fifteen, Gauss attended college at Brunswick,

through the help of the Duke of Brunswick. The Duke later also

supported the young mathematician in enrolling at the renowned

university at Gottingen. There, on March 30, 1796, Gauss wrote

the first page in his famous diary. The diary had only nineteen

pages, but in these pages Gauss recorded 146 brief statements of

important and powerful mathematical results he had derived. It was

later discovered that almost every important mathematical idea

published by any mathematician in the late eighteenth and in the

nineteenth centuries had been preceded by one of the entries in

Gauss' unpublished diary. The diary remained hidden until it was

found in the possession of Gauss' grandson in Hamlin in 1898.

Gauss' results in number theory, which were shared with the

mathematicians of his day by regular correspondence, were of

great importance in all attempts by mathematicians to prove

Fermat's Last Theorem. Many of these results were contained in a

book on number theory which Gauss published in Latin in 1801,

when he was 24. The book, Distfuisitiones Arithmeticae, was

translated into French and published in Paris in 1807 and received

much attention. It was recognized as the work of genius. Gauss

dedicated it to his patron, the Duke of Brunswick.

Gauss was a distinguished scholar of classical languages as well.

On entering college he was already a master of Latin, and his

interest in philology precipitated a crisis in his career. Should he

follow the study of languages or of mathematics? The turning point

was March 30, 1796. From his diary, we know that on that day

the young Gauss decided definitely to specialize in mathematics. In

mathematics and statistics (where he is credited with the ingenious

method of least squares for finding a line to fit a data set) he

contributed to many areas, but he believed that number theory was

the heart of all of mathematics.

But why did the world's greatest mathematical genius never try

to prove Fermat's Last Theorem? Gauss' friend H. W. M. Olbers

wrote him a letter from Bremen on March 7, 1816, in which he

told Gauss that the Paris Academy offered a big prize for anyone

who would present a proof or a disproof of Fermat's Last Theorem.

Gauss surely could use the money, his friend suggested. At that

time, as he did throughout his mathematical career, Gauss received

financial support from the Duke of Brunswick, which allowed him to

do his mathematical work without the need for additional

employment. But he was far from rich. And, as Olbers suggested, no

other mathematician had nearly his expertise or ability. "It seems

right to me, dear Gauss, that you should get busy about this," he

concluded.

But Gauss would not be tempted. Possibly he knew how

deceptive Fermat's Last Theorem really was. The great genius in

number theory may have been the only mathematician in Europe to

realize just how difficult it would be to prove. Two weeks later,

Gauss wrote to Olbers his opinion of Fermat's Last Theorem: "I am

very much obliged for your news concerning the Paris prize. But I

confess that Fermat's Theorem as an isolated proposition has very

little interest for me, because 1 could easily lay down a multitude

of such propositions, which one could neither prove nor dispose of."

Ironically, Gauss made great contributions to the branch of

mathematics known as complex analysis—an area incorporating the

imaginary numbers worked on by Euler. Imaginary numbers would

have a decisive role in twentieth-century understanding of the con-

text of Fermat's Last Theorem.

Imaginary Numbers

The complex number field is a field of numbers based on the usual

real numbers, and what are called imaginary numbers, which were

known to Euler. These numbers arose when mathematicians were

looking to define as a number the solution of an equation such as x2

+ 1 = 0. There is no "real" solution to this simple equation,

because there is no real number which, when squared, gives -1—

the number that when added to 1 will give the answer zero. But if

we could somehow define the square root of negative one as a

number, then—while not a real number—it would be the solution to

the equation.

The number line was therefore extended to include imaginary

numbers. These numbers are multiples of the square root

of - 1 , denoted by i. They were put on their own number line,

perpendicular to the real number line. Together, these two axes

give us the complex plane. The complex plane is shown below. It

has many surprising properties, such as rotation being

multiplication by i.

Multiplication by i rotates counterclockwise.

The complex plane is the smallest field of numbers that contains

the solutions of all quadratic equations. It was found to be very

useful, even in applications in engineering, fluid mechanics, and

other areas. In 1 8 1 1 , decades ahead of his time, Gauss was

studying the behavior of functions on the complex plane. He

discovered some amazing properties of these functions, known as

analytic functions. Gauss found that analytic functions had special

smoothness, and they allowed

for particularly neat calculations. Analytic functions preserved

angles between lines and arcs on the plane—an aspect that would

become significant in the twentieth century. Some analytic

functions, called modular forms, would prove crucial in new

approaches to the Fermat problem.

In his modesty, Gauss did not publish these impressive results.

He wrote of them in a letter to his friend Friedrich Wilhelm Bessel

(1784-1846). When the theory reemerged without Gauss' name

attached to it years later, other mathematicians were given credit

for work on the same analytic functions Gauss understood so well.

Sophie Germain

One day Gauss received a letter from a certain "Monsieur Leblanc."

Leblanc was fascinated with Gauss' book, Discjuisitiones

Arithmeticae, and sent Gauss some new results in arithmetic the-

ory. Through the ensuing correspondence on mathematical

matters, Gauss gained much respect for Mr. Leblanc and his work.

This appreciation did not diminish when Gauss discovered that

neither was his correspondent's real name Leblanc, nor was the

writer of the letters a "Mr." The mathematician writing so eloquently

to Gauss was one of very few women active in the profession at

that time, Sophie Germain (1776-1831). In fact, upon

discovering the deception, Gauss wrote her:

But how to describe to you my admiration and

astonishment at seeing my esteemed

correspondent Mr. Leblanc metamorphose himself

into this illustrious personage who gives such a

brilliant example of what 1 would find difficult to

believe...

(A letter from Gauss to Sophie Germain, written from Brunswick on

Gauss' birthday, as stated in French at the end of his letter:

"Bronsvic ce 30 avril 1807 jour de ma naissance.")

Sophie Germain assumed a man's name to avoid the prejudice

against women scientists prevalent in those days and to gain

Gauss' serious attention. She was one of the most important

mathematicians to attempt a proof of Fermat's Last Theorem, and

to make considerable headway on the problem. Sophie Germain's

Theorem, which gained her much recognition, states that if a

solution of Fermat's equation with «=5 existed, then all three

numbers had to be divisible by 5. /The theorem separated

Fermat's Last Theorem into two cases: Case I for numbers not

divisible by 5, and Case II for numbers that are. The theorem was

generalized to other powers, and Sophie Germain gave a general

theorem which allowed for a proof of Fermat's Last Theorem for all

prime numbers, n, less than 100 in Case I. This was an important

result, which reduced the possible cases where Fermat's Last

Theorem might fail for primes less than 100 to only Case II.10

Sophie Germain had to drop the disguise she was using when

Gauss asked his friend "Leblanc" for a favor. In 1807, Napoleon

was occupying Germany. The French were imposing war fines on

the Germans, and were determining the amounts owed by each

resident based on what they perceived each person was worth. As

a prominent professor and astronomer at Gottingen, Gauss was

determined to owe 2,000 francs—far beyond his means. A number

of French mathematicians who were friends of the great Gauss

offered their help, but he refused to take their money. Gauss

wanted some

one to intercede on his behalf with the French General Per-nety in

Hanover.

He wrote his friend, Monsieur Leblanc, to ask if Leblanc might

contact the French General on Gauss' behalf. When Sophie Germain

gladly complied, it became clear who she was. But Gauss was

thrilled, as seen from his letter, and their correspondence continued

and developed further on many mathematical topics. Unfortunately,

the two never met. Sophie Germain died in Paris in 1831, before

the University of Gottingen could award her the honorary doctorate

which Gauss had recommended she receive.

Sophie Germain had many other achievements to her credit in

addition to her contributions to the solution of Fermat's Last

Theorem. She was active in the mathematical theories of acoustics

and elasticity, and other areas in applied and pure mathematics. In

number theory, she also proved theorems on which prime numbers

can lead to solvable equations.

The Blazing Comet of i 8 a

Gauss did much important work in astronomy, determining the

orbits of planets. On August 22, 1811, he first observed a comet

barely visible in the night sky. He was able to predict the comet's

precise trajectory toward the sun. When the comet became clearly

visible and blazed across the sky the superstitious and oppressed

peoples of Europe saw in it a sign from heaven signifying the

coming demise of Napoleon. Gauss saw in the comet a realization

of the orbit he had predicted for it to exact numerical accuracy. But

the unscientific masses were also right—the next year Napoleon

was defeated and

retreated from Russia. Gauss was amused. He was not unhappy to

see the Emperor defeated after French forces had extorted such

high amounts from him and his countrymen.

The Disciple

The Norwegian mathematician Niels Henrik Abel visited Paris in

October of 1826. There, he tried to meet other mathematicians—

Paris was at that time a mecca for mathematics. One of the people

who impressed Abel the most was Peter Gustav Lejeune Dirichlet

(1805-1859), a Prussian who was also visiting Paris and

gravitated toward the young Norwegian, at first thinking he was a

fellow Prussian. Abel was much impressed by the fact that Dirichlet

had proved Fermat's Last Theorem for M=5. He wrote about it in a

letter to a friend, mentioning that this was also done by Adrien-

Marie Legendre (1752-1833). Abel described Legendre as

extremely polite but very old. Legendre had proved the Fermat

result for n = 5 independently of Dirichlet, two years after him.

Unfortunately, this was always happening to Legendre—much of his

work was superseded by that of younger mathematicians.

Dirichlet was a friend and disciple of Gauss. When Gauss' great

book, the Discfuisitiones Arithmeticae, was published, it quickly

went out of print. Even mathematicians whose work was related to

that of Gauss could not obtain a copy. And many who did, did not

understand the depth of Gauss' work. Dirichlet had his own copy.

He carried this copy of Gauss' book with him on his many travels, to

Paris, Rome, and elsewhere on the Continent. Everywhere he went,

Dirichlet slept with the book under his pillow. Gauss' book became

known as

the book of the seven seals: The gifted Dirichlet is known as the

person who broke the seven seals. Dirichlet did more than anyone

else to explain and interpret the book of his great master to the rest

of the world.

Besides amplifying and explaining the DisQuisitiones, as well as

proving Fermat's Last Theorem for the power five, Dirichlet did

other great mathematics. One interesting result that Dirichlet

proved was connected with the following progression of numbers:

a, a + b, a + lb, a + 3b, a + 4b,... and so on, where the

numbers a and b are integers that have no common divisor other

than 1 (that is, they are numbers such as 1 and 3, or 3 and 5,-

rather than numbers such as 1 and 4, which have the common

divisor 1, or the numbers 6 and 9, which have the common

divisor 3). Dirichlet proved that this progression of numbers

contains infinitely many prime numbers. The amazing element of

Dirichlet's proof was that he had crafted it using an area of

mathematics which seemed in those days to be very far from

number theory, where this problem rightfully belongs. In his proof,

Dirichlet used the field called analysis, an important area of

mathematics which contains the calculus. Analysis deals with

continuous things: functions on a continuum of numbers on the line,

which seems very far from the discrete world of integers and prime

numbers—the realm of number theory.

It would be a similar bridge across seemingly different branches

of mathematics that would usher in the modern philosophy solving

the Fermat mystery in our own century. Dirichlet was a bold pioneer

in this area of unifying disparate branches of mathematics. The

student later inherited the mas

ter's position. When Gauss died in 1855, Dirichlet left his

prestigious job in Berlin to accept the honor of replacing Gauss at

Gottingen.

Napoleon's Mathematicians

The Emperor of the French loved mathematicians, even if he wasn't

one himself. Two who were especially close to him were

GaspardMonge (1746-1818), and Joseph Fourier (1768-1830).

In 1798, Napoleon took the two mathematicians with him to Egypt,

to help him "civilize" that ancient country.

Fourier was born in Auxerre, France, on March 21, 1768, but at

the age of eight he was orphaned and was helped by the local

Bishop to enter the military school. Even at the age of twelve,

Fourier showed great promise and was writing sermons for church

dignitaries in Paris, which they passed off as their own. The French

Revolution of 1789 saved the young Fourier from life as a priest.

Instead he became a professor of mathematics, and an enthusiastic

supporter of the revolution. When the revolution gave way to the

Terror, Fourier was repulsed by its brutality. He used his eloquence,

developed over years of writing sermons for others, to preach

against the excesses. Fourier also used his great public speaking

skills in teaching mathematics at the best schools in Paris.

Fourier was interested in engineering, applied mathematics, and

physics. At the Ecole Polytechnique, he did serious research in

these areas, and many of his papers were presented to the

Academy. His reputation brought him to the attention of Napoleon,

and in 1798 the Emperor asked Fourier to accompany him, aboard

his flagship, along with the French fleet of five hun-

dred ships headed for Egypt. Fourier was to be part of the Legion of

Culture. The Legion's charge was to "endow the people of Egypt

with all the benefits of European civilization." Culture was to be

brought to these people while they were being conquered by the

invading armada. In Egypt, the two mathematicians founded the

Egyptian Institute, and Fourier stayed there until 1802, when he

returned to France and was made a prefect of the region around

Grenoble. There he was responsible for many good public works

such as draining marshlands and stamping out malaria. With all that

work, Fourier, the mathematician-turned-administrator, managed to

find time to do his best mathematical thinking. Fourier's

masterpiece was the mathematical theory of heat, answering the

important question: How is heat conducted? This work won him a

Grand Prize from the Academy in 1812. Part of his work was based

on experiments he had done in the deserts of Egypt during his

years there. Some of his friends believed that these experiments,

including putting himself through the intense heat that developed in

closed rooms, contributed to his early death at the age of 62.

Fourier spent the last years of his life telling tales about

Napoleon and his close association with him, both in Egypt and after

Napoleon's escape from Elba. Fourier's research on heat, however,

was what immortalized him, for he developed an important theory

of periodic functions. A series of such periodic functions, when used

in a particular way to estimate another function, is called a Fourier

Series.

AMIR D . ACZEL

Periodic Functions

The clearest example of a periodic function is your watch. Minute by

minute, the large hand moves around a circle, and after sixty

minutes it comes back to the exact same place where it started.

Then it continues and in exactly sixty minutes it again returns to the

same spot. (Of course the little hand will have changed positions as

the hours go by.) The minutes hand on a watch is a periodic

function. Its period is exactly sixty minutes. In a sense, the space of

all minutes of eternity—the set of infinitely many minutes from now

until forever—can be wrapped by the big hand of a watch on the

outer edge of the face of the watch:

To take another example, on a locomotive speeding on the railroad

track the arm transferring power from the engine to the wheel

moves up and down along that wheel as it rotates. Every full circle

of the wheel, the arm comes back to its original position—this arm,

too, is periodical. The vertical height of the arm, when the radius of

the train wheel is one unit, is

defined as the sine function. This is the elementary

trigonometric function taught at school. The cosine

function is the horizontal measure of the arm. Both

sine and cosine are functions of the angle the arm

makes with a horizontal line through the center of

the wheel. This is shown below.

As the train moves forward, the vertical height of the arm traces

a wave-pattern as seen above. This pattern is periodic. Its period is

360 degrees. First the arm's height is zero, then it goes up in a

wave-like fashion until it reaches one, then it decreases, then it

goes to zero again, then to negative values until one, then its

negative magnitude decreases to zero. And then the cycles starts

all over again.

What Fourier discovered was that most functions can be

estimated to any degree of accuracy by the sum of many (the-

oretically infinitely many, for near-perfect accuracy) sine and cosine

functions. This is the famous result of the Fourier series. This

expansion of any function in terms of the sum of many sine and

cosine functions is useful in many applications of mathematics

when the actual mathematical expression of interest is difficult to

study but the sum of the sines and cosines, all multiplied by

different factors, can be easily

manipulated and evaluated—and this is especially

practical on a computer. The area of mathematics

known as numerical analysis is concerned with

computer methods for evaluating functions and other

numerical quantities. Fourier analysis is a substantial

part of numerical analysis, and consists of techniques

for studying difficult problems, many with no closed-

form (that is, given by a simple mathematical

expression) solution, using the Fourier series of

periodic function. After Fourier's pioneering work,

expansions using other simple functions, mostly

polynomials (that is, increasing powers of a variable:

squares, cubes, and so on) were also developed.

When your calculator computes the square root of a

number, it is doing so by an approximation based on

such a method. The Fourier series of sines and

cosines is especially useful in estimating phenomena

that are naturally the sum of periodic elements—for

example, music. A musical piece may be decomposed

into its harmonics. The tides, the phases of the

moon, and sun spots are examples of simple periodic

phenomena.

While the applications of Fourier's periodic

functions to natural effects and computational

methods are of great importance, the surprising fact

is that Fourier series and analysis found some useful

applications in pure mathematics, a field that was

never one of Fourier's main interests. In the

twentieth century, Fourier series found a role in

number theory as the tool for transforming

mathematical elements from one area to another in

the work of Goro Shimura. (The proof of Shimura's

conjecture was at the crux of proving Fermat's Last

Theorem.) Extensions of the periodic functions of

Fourier to the complex plane, linking together these

two areas of mathematics, would

lead to the discovery of automorphic functions and

modular forms—which also had a crucial impact on

Fermat's Last Theorem through the early twentieth

century work of another French mathematician, Henri

Poincare.

A Lame Proof

At the March 1, 1847 meeting of the Paris Academy,

the mathematician Gabriel Lame (1795-1870)

announced very excitedly that he had obtained a

general proof of Fermat's Last Theorem. Until then,

only specific powers, n, were attacked, and a proof

had been given of the theorem in the cases n = 3, 4,

5, 7. Lame suggested that he had a general approach

to the problem, which would work for any power n.

Lame's method was to factor the left side of Fermat's

equation, xn + yn, into linear factors using complex

numbers. Lame then went on to state modestly that

the glory should not all go to him, since the method

he suggested was introduced to him by Joseph

Liouville (1809-1882). But Liouville took the podium

after Lame and brushed aside any praise. Lame had

not proved Fermat's Last Theorem, he quietly stated,

because the factorization he suggested was not

unique (that is, there were many ways to carry out

the factorization, so there was no solution). It was a

gallant attempt, one of many, but it did not bear

fruit. However, the idea of factorization, that is,

breaking down the equation to a product of factors,

was to be tried again.

Ideal Numbers

The person to again try factorization was Ernst

Eduard Kum-mer (1810-1893)—the man who got

closer than anyone else in

his time to a general solution of Fermat's problem.

Kummer, in fact, invented an entire theory in

mathematics, the theory of ideal numbers, in

attempting to prove Fermat's Last Theorem.

Kummer's mother, a widow when her son was only

three, worked hard to assure her boy a good

education. At the age of eighteen he entered the

University of Halle, in Germany, to study theology

and prepare himself for life in the church. An

insightful professor of mathematics with enthusiasm

for algebra and number theory got the young

Kummer interested in these areas, and he soon

abandoned theology for mathematics. In his third

year as a student, the young Kummer solved a dif-

ficult problem in mathematics for which a prize had

been offered. Following this success, he was awarded

a doctorate in mathematics at the age of twenty-one.

But Kummer was unable to find a university

position and therefore had to take a teaching job at

his old gymnasium (high school). He remained a

schoolteacher for another ten years. Throughout this

period, Kummer did much research, which he

published and wrote in letters to several leading

mathematicians of his time. His friends realized how

sad it was to have such a gifted mathematician spend

his life teaching high school math. With the help of

some eminent mathematicians, Kummer was given

the position of professor at the University of Breslau.

A year later, in 1855, Gauss died. Dirichlet had taken

Gauss' place at Gottingen, leaving behind his old job

at the prestigious University of Berlin. Kummer was

chosen to replace Dirichlet in Berlin. He remained in

that position until his retirement.

Kummer worked on a wide range of problems in

mathemat-

fi7

ics, ranging from very abstract to very applied—even

applications of mathematics in warfare. But he made

his name for his extensive work on Fermat's Last

Theorem. The French mathematician Augustin-Louis

Cauchy (1789-1857), thought on a number of

occasions that he had found a general solution to the

Fermat problem. But the restless, careless Cauchy

realized in every attempt that the problem was much

bigger than he had assumed. The fields of numbers

he was working with always failed to have the

property he required. Cauchy left the problem and

went to work on other things.

Kummer became obsessed with Fermat's Last

Theorem, and his attempts at a solution led him

down the same futile track taken by Cauchy. But

instead of giving up hope when he recognized that

the number fields involved failed to have some prop-

erty, he instead invented new numbers that would

have the property he needed. These numbers he

called "ideal numbers." Kummer thus developed an

entirely new theory from scratch, which he used in

his attempts to prove Fermat's Last Theorem. At one

point, Kummer thought he finally had a general

proof, but this unfortunately fell short of what was

needed.

Nonetheless, Kummer did achieve tremendous

gains in his attack on the Fermat problem. His work

with ideal numbers enabled him to prove Fermat's

Last Theorem for a very extensive class of prime

numbers as the exponent n. Thus, he was able to

prove that Fermat's Last Theorem was true for an

infinite number of exponents, namely those that are

divisible by "regular" primes. The "irregular" prime

numbers eluded him. The only irregular primes less

than 100 are 37, 59, and 67. Kummer then worked

separately on the problem of these

irregular prime numbers and was eventually able to

prove Fermat's theorem for these numbers. By the

1850s, using Kum-mer's incredible breakthroughs,

Fermat's Last Theorem was known to be true for all

exponents less than n = 100, as well as infinitely

many multiples of prime numbers in this range. This

was quite an achievement, even if it was not a

general proof and still left out infinitely many

numbers for which it was not known whether the

theorem held true.

In 1816, the French Academy of Sciences offered a

prize to anyone who would prove Fermat's Last

Theorem. In 1850 the Academy again offered a gold

medal and 3,000 francs to the mathematician who

would prove Fermat's Last Theorem. In 1856, the

Academy decided to withdraw the award, since it

didn't seem like a solution to Fermat's problem was

imminent. Instead, the Academy decided to give the

award to Ernst Eduard Kummer, "for his beautiful

researches on the complex numbers composed of

roots of unity and integers." Thus Kummer was

awarded a prize for which he never even applied.

Kummer continued his tireless efforts on Fermat's

Last Theorem, stopping his research only in 1874.

Kummer also did pioneering work on the geometry of

four-dimensional space. Some of his results are

useful in the field of modern physics known as

quantum mechanics. Kummer died of the flu in his

eighties, in 1893.

Kummer's success with ideal numbers is even

more highly praised by mathematicians than the

actual advances he made in the solution of Fermat's

problem using these numbers. The fact that this

notable theory was inspired by attempts to solve Fer-

mat's Last Theorem shows how new theories can be

developed

by attempts to solve particular problems. In fact,

Kummer's theory of ideal numbers led to what are

now known as "ideals," which have had an impact on

the work of Wiles and other mathematicians on

Fermat's theorem in the twentieth century.

Another Prize

In 1908, the Wolfskehl Prize of one hundred thousand

marks was offered in Germany for anyone who would

come up with a general proof of Fermat's Last

Theorem. In the first year of the prize, 621

"solutions" were submitted. All of them were found to

be false. In the following years, thousands more

"solutions" were submitted, with the same effect. In

the 1920s, German hyperinflation reduced the real

value of the 100,000 marks to nothing. But false

proofs of Fermat's Last Theorem continued to pour

in.

Geometry Without Euclid

New developments started taking place in

mathematics in the nineteenth century. Janos Bolyai

(1802-1860) and Nicolas Ivanovitch Lobachevsky

(1793-1856), a Hungarian and a Russian respectively,

changed the face of geometry. By doing away with

Euclid's axiom that two parallel lines never meet,

these two were able independently to formulate a

geometrical universe which maintained many of

Euclid's properties but allowed two parallel lines to

meet at a point at infinity. A different geometry can

be seen, for example, on a sphere such as the globe.

Locally, two longitude lines are parallel. But in

reality, as they are followed to the North Pole, the

two lines meet there. New geome

tries solve many problems and explain situations

which, until that time, seemed mysterious and

without a solution.

Beauty and Tragedy —

Abstract algebra, a field derived from the familiar

algebra taught in school as a system for solving

equations, was developed in the nineteenth century.

In this area, the beautiful theory of Galois stands out.

Evariste Galois was born in 1811 in the small

village of Bourg-la-Reine, outside Paris. His father

was the mayor of the town and a staunch republican.

Young Evariste grew up on the ideals of democracy

and freedom. Unfortunately, most of France was at

that time heading in the opposite direction. The

French Revolution had come and gone, and so had

Napoleon. But the dreams of freedom, equality, and

fraternity had not yet been achieved. And the

royalists were enjoying a comeback in France, with a

Bourbon once again crowned King of the French—now

to rule together with the people's representatives.

Evariste's life was steeped in the lofty ideals of the

revolution. He was a good ideologue, and he gave

some moving speeches to the republicans. As a

mathematician, on the other hand, he was a genius

of unparalleled ability. As a teenager, Galois

absorbed the entire theories of algebra and

equations known to accomplished mathematicians of

his day, and—while still a schoolboy—developed his

own complete system, known today as Galois Theory.

Unfortunately, he was not to enjoy any recognition in

his tragically short life. Galois stayed up nights at his

boarding school, while everyone else slept, and

wrote down his theory. He sent it to the head of the

French Academy of Science—the mathematician

Cauchy—hoping that Cauchy would help him publish

the theory. But Cauchy was not only very busy, he

was also arrogant and careless. And Galois' brilliant

manuscript ended up in a trash can, unread.

Galois tried again, with the same result. In the

meantime, he failed to pass the entry exams to the

Lxole Polytechnique, where most of the celebrated

mathematicians of France got their training. Galois

had a habit of always working things out in his head.

He never took notes or wrote things down until he

had actual results. This method concentrated on ideas

rather than detail. Young Evariste had little patience

for, or interest in, the details. It was the great idea,

the beauty of the larger theory, that interested him.

Consequently, Galois was not at his best when taking

an examination in front of a blackboard. And this is

what caused him to fail twice in his attempts to gain

entrance to the school of his dreams. Twice in front

of the blackboard he did not perform well writing

things down, and he got irritated when asked for

details he just didn't consider

important. It was a tragedy that an

incrediblyintelligent young person would be

questioned by much less able examiners who could

not understand his deep ideas and took his

reluctance to give trivial details for ignorance. When

he realized he was going to fail the second and last

permissible attempt, and that the gates of the school

would be closed to him forever, Galois threw the

blackboard eraser in the face of his examiner.

Galois had to make do with a second-best choice,

the Ecole Normale. But even there he did not fare

well. Galois' father, the mayor of Bourg-la-Reine, was

the target of clerical intrigues in the town. An

unscrupulous priest circulated pornographic verses,

signing them with the mayor's name. Months of such

persecution caused Galois' father to lose his

confidence and he became convinced that the world

was out to get him. Slowly losing touch with reality,

he went to Paris. There, in an apartment not far from

where his son was studying, the father killed himself.

The young Galois never recovered from this tragedy.

Obsessed with the lost cause of the 1830 revolution,

and frustrated with the school director, whom he

considered an apologist for the royalists and clerics,

Galois wrote a scathing letter criticizing the director.

He was inspired to do so after three days of rioting in

the streets, when students all over Paris were

revolting against the regime. Galois and his

classmates were kept locked in their school, unable

to scale the tall fence. The angry Galois sent his

blistering letter criticizing the school director to the

Gazette des ficoles. As a result, he was expelled from the

school. But Galois was undaunted—he wrote a second

letter to the Gazette, and addressed the students at

the school to speak up for honor and conscience. He

got no response.

Out of school, Galois started offering private

lessons in mathematics. He wanted to teach his own

mathematical theories, outside the French schools,

when he was all of nineteen years old. But Galois

could not find students to teach—his theories were

too advanced; he was far ahead of his time.

Facing an uncertain future and doomed not to be

able to pursue a decent education, in desperation

Galois joined the artillery branch of the French

National Guard. Among the National Guard, headed

in its past by Lafayette, there were many liberal

elements close to young Galois' political philosophy.

While in the Guard, Galois made one last attempt to

publish his mathematical work. He wrote a paper on

the general solution of equations—today recognized

as the beautiful Galois Theory—and sent it to Simeon-

Denis Poisson (1781-1840) at the French Academy of

Sciences. Poisson read the paper but determined

that it was "incomprehensible." Once again, the

nineteen-year-old was so far ahead of any of the

older French mathematicians of his day that his

elegant new theories went way over their heads. At

that moment, Galois decided to abandon

mathematics and to become a full-time revolutionary.

He said that if a body was needed to get the people

involved with the revolution, he would donate his.

On May 9, 1831, two hundred young republicans

held a banquet in which they protested against the

royal order disbanding the artillery of the National

Guard. Toasts were offered to the French Revolution

and its heroes, as well as to the new revolution of

1830. Galois stood up and proposed a toast. He said

"To Louis Philippe," the Duke of Orleans, who was

now King of the French. While saying this, holding up

his

glass, Galois was also holding up an open pocket-

knife in his other hand. This was interpreted as a

threat on the life of the King, and caused a riot. The

next day Galois was arrested.

In his trial for threatening the life of the King,

Galois' attorney claimed that Galois had actually

said, while holding his knife, "To Louis Philippe, if he

becomes a traitor." Some of Galois' artillery friends

who were present testified to this, and the jury found

him not guilty. Galois retrieved his knife from the

evidence table, closed it and put it in his pocket, and

left the courtroom a free man. But he was not free

for very long. A month later he was arrested as a

dangerous republican, and kept in jail without a

charge while the authorities looked for a charge

against him that would stick. They finally found one—

wearing the uniform of the disbanded artillery. Galois

was tried for this/:harge and sentenced to six months

in jail. The royalists wercpleased to finally put away a

twenty-year-old they considered to be a dangerous

enemy of the regime. Galois was paroled after some

time and was moved to a halfway facility. What

happened next is open to question. While on parole,

Galois met a young woman and fell in love. Some

believe he was set up by his royalist enemies who

wanted to put an end to his revolutionary activities

once and for all. At any rate, the woman with whom

he got involved was of questionable virtue ("une

cocjuette de has e'tage"). As soon as the two became

lovers, a royalist came to "save her honor" and

challenged Galois to a duel. The young

mathematician was left no way out of the mess. He

tried everything he could to talk the man out of the

duel, but to no avail.

The night before the duel, Galois wrote several

letters.

These letters to his friends lend support to the

theory that he was framed by the royalists. He wrote

that he was challenged by two of the royalists and

that they had put him on his honor not to tell his

republican friends about the duel. "I die the victim of

an infamous coquette. It is in a miserable brawl that

my life is extinguished. Oh, why die for so trivial a

thing, why die for something so despicable!" But

most of that last night before the duel, Galois

carefully put down on paper his entire mathematical

theory, and sent it to his friend Auguste Chevalier. At

dawn on May 30, 1832, Galois faced his challenger on

a deserted field. He was shot in the stomach and left

lying in agony alone in the field. No one bothered to

call a doctor. Finally a peasant found him and

brought him to the hospital, where he died the next

morning. He was twenty years old. In 1846, the

mathematician Joseph Liouville edited and published

Galois' elegant mathematical theory in a journal.

Galois' theory would supply the crucial step in the

method used a century and a half later in attacking

Fermat's Last Theorem.

Another Victim

Cauchy's carelessness and arrogance ruined the life

of at least one other brilliant mathematician. Niels

Henrik Abel (1802-1829) was the son of the pastor of

the village of Findo in Norway. When he was sixteen,

a teacher encouraged Abel to read Gauss' famous

book, the Discjuisitiones. Abel was even successful in

filling in some gaps in the proofs of some of the

theorems. But two years later, his father died and

young Abel had to postpone his study of mathematics

and concentrate his efforts on supporting the family.

In spite of the great difficulties he

faced, Abel managed to continue some study of

mathematics and made a remarkable mathematical

discovery at the age of nineteen. He published a

paper in 1824 in which he proved that no solution

was possible for equations of the fifth degree. Abel

had thus solved one of the most celebrated problems

of his day. Yet the gifted young mathematician had

still not been offered an academic position, which he

badly needed to support his family and so he sent his

work to Cauchy for evaluation and possible

publication and recognition. The .paper Abel sent

Cauchy was of extraordinary power and generality.

But Cauchy lost it. When it appeared in print, years

later, it was much too late to help Abel. In 1829 Abel

died from tuberculosis, brought on by poverty and

the strain of supporting his family in dire

circumstances. Two days after his death, a letter

arrived addressed to Abel informing him he had been

appointed professor at the University of Berlin.

The concept of an Abelian Group (now considered a

word and written with a small "a," abelian) is very

important in modern algebra and is a crucial element

in the modern treatment of the Fermat problem. An

abelian group is one where the order of mathematical

operations can be reversed without affecting the

outcome. An abelian variety is an even more abstract

algebraic entity, and its use was also important in

modern approaches to the solution of Fermat's Last

Theorem.

D ed ek in d 's Idea ls

The legacy of Carl Friedrich Gauss continued through

the centuries. One of Gauss' most notable

mathematical successors was Richard Dedekind

(1831-1916), born in the same

town as the great master, in Brunswick, Germany.

Unlike Gauss, however, as a child Dedekind showed

no great interest in or capacity for mathematics. He

was more interested in physics and chemistry, and

saw mathematics as a servant to the sciences. But at

the age of seventeen, Dedekind entered the same

school where the great Gauss got his mathematical

training—Caroline College—and there his future

changed. Dedekind became interested in

mathematics and he pursued that interest in

Gottingen, where Gauss was teaching. In 1852, at the

age of 21, Richard Dedekind received his doctorate

from Gauss. The master found his pupil's dissertation

on the calculus "completely satisfying." This was not

such a great compliment, and in fact Dedekind's

genius had not yet begun to manifest itself.

In 1854, Dedekind was appointed a lecturer in

Gottingen. When Gauss died in 1855 and Dirichlet

moved from Berlin to take his position, Dedekind

attended all of Dirichlet's lectures at Gottingen and

edited the latter's pioneering treatise on number

theory, adding a supplement based on his own work.

This supplement contained an outline of the theory

Dedekind developed for algebraic numbers, which

are defined as solutions of algebraic equations. They

contain multiples of square roots of numbers along

with rational numbers. Algebraic number fields are

very important in the study of Fermat's equation, as

they arise from the solution of various kinds of

equations. Dedekind thus developed a significant

area within number theory.

Dedekind's greatest contribution to the modern

approach to Fermat's last Theorem was his

development of the theory of ideals, abstractions of

Kummer's ideal numbers. A century

after their development by Dedekind, ideals would

inspire Barry Mazur, and Mazur's own work would be

exploited by Andrew Wiles.

In the 1857-8 academic year, Richard Dedekind

gave the first mathematics course on Galois theory.

Dedekind's understanding of mathematics was very

abstract, and he elevated the theory of groups to the

modern level at which it is understood and taught

today. Abstraction made the twentieth-century

approach to Fermat's problem possible. Dedekind's

groundbreaking course on the theories developed by

Galois was a great step in this direction. The course

was attended by two students.

Then Dedekind's career took a strange turn. He left

Gottingen for a position in Zurich, and after five

years, in 1862, he returned to Brunswick, where he

taught at a high school for fifty years. No dne has

been able to explain why a brilliant mathematician

who brought algebra to an incredibly high level of

abstraction and generality suddenly left one of the

most prestigious positions at any European

university to teach at an unknown high school.

Dedekind never married and lived for many years

with his sister. He died in 1916, and maintained a

sharp, active mind to his last day.

Fin de Siecle

At the turn of the nineteenth century, there lived in

France a mathematician of great ability in a

surprisingly wide variety of areas. The breadth of

knowledge of Henri Poincare (1854-1912) extended

beyond mathematics. In 1902 and later, when he was

already a renowned mathematician, Poincare wrote

popular

books on mathematics. These paperbacks, read by

people of all ages, were a common sight in the cafes

and the parks of Paris.

Poincare was born to a family of great achievers.

His cousin, Raymond Poincare, rose to be the

president of France during the First World War.

Other family members held government and public

service positions in France as well.

From a young age, Henri displayed a powerful

memory. He could recite from any page of a book he

read. His absentmind-edness, however, was

legendary. A Finnish mathematician once came all

the way to Paris to meet with Poincare and discuss

some mathematical problems. The visitor was kept

waiting for three hours outside Poincare's study

while the absentminded mathematician paced back

and forth—as was his habit throughout his working

life. Finally, Poincare popped his head into the

waiting room and exclaimed: "Sir, you are disturbing

me!" upon which the visitor summarily left, never to

be seen in Paris again.

Poincare's brilliance was recognized when he was

in elementary school. But since he was such a

universalist—a renaissance man in the making—his

special aptitude for mathematics did not yet

manifest itself. He distinguished himself with his

excellence in writing at an early age. A teacher who

recognized and encouraged his ability treasured his

school papers. At some point, however, the

concerned teacher had to caution the young genius:

"Don't do so well, please...try to be more ordinary."

The teacher had good reason to make this sugges-

tion. Apparently, French educators had learned

something from the misfortunes of Galois half a

century earlier—teachers found that gifted students

often failed at the hands of uninspired examiners.

His teacher was genuinely worried that

Poincare was so brilliant that he might fail those

exams. As a child, Poincare was already

absentminded. He often skipped meals because he

forgot whether or not he had eaten.

Young Poincare was interested in tjie classics and

he learned to write well. As a teenager he became

interested in mathematics and immediately excelled

in it. He would work out problems entirely in his head

while pacing in a room—then sit down and write

everything very hastily. In this he resembled Galois

and Euler. When Poincare finally took his exams, he

almost failed in math, as his elementary school

teacher had feared years earlier. But he did pass,

only because—at seventeen—his renown as a

mathematician was so great that the examiners

didn't dare fail him. "Any student other than Poincare

would have been given a failing grade," the chief

examiner declared as he passed the student who

went on to the Ecole Polytechnique and became the

greatest French mathematician of his time.

Poincare wrote scores of books on mathematics,

mathematical physics, astronomy, and popular

science. He wrote research papers of over five

hundred pages on new mathematical topics he

developed. He made major contributions to topology,

the area started by Euler. However, Poincare's

results were so important that this branch of

mathematics is considered to have truly been

launched only in 1895, with the publication of

Poincare's Analysis Situs. Topology—the study of shapes

and surfaces and continuous functions—was

important in understanding Fermat's problem in the

late twentieth century. But even more essential to

the modern approach to Fermat's Last Theorem was

another area pioneered by Henri Poincare.

FERMAT'S LAST THEOREM

Modular Forms

Poincare studied periodic functions, such as the sines

and cosines of Fourier—not on the number line as

Fourier had done, but in the complex plane. The sine

function, sin x, is the vertical height on a circle with

radius 1 when the angle is x. This function is periodic:

it repeats itself over and over every time the angle

completes a multiple of its period, 360 degrees. This

periodicity is a symmetry. Poincare examined the

complex plane, which contains real numbers on the

horizontal axis and imaginary ones on the vertical one

as shown below.

Here, a periodic function could be conceived as having a periodicity

both along the real axis and along the imaginary axis. Poincare went

even further and posited the existence of functions with a wider

array of symmetries. These were functions that remained

unchanged when the complex variable z was changed

according to/(z) >j(az+b/cz+d). Here the elements a, b, c,

d, arranged as a matrix, formed an algebraic group. This means

t h a t t h e r e a r e i n f i n i t e l y m a n y p o s s i b l e v a r i a t i o n s . T h e y a l l c o m m u t e w i t h e a c h o t h e r a n d t h e f u n c t i o n / i s invariant u n d e r t h i s g r o u p o f t r a n s f o r m a t i o n s . P o i n c a r e c a l l e d s u c h w e i r d f u n c t i o n s a u t o m o r p h i c f o r m s .

T h e a u t o m o r p h i c f o r m s w e r e v e r y , v e r y s t r a n g e c r e a t u r e s s i n c e t h e y s a t i s f i e d m a n y i n t e r n a l s y m m e t r i e s . P o i n c a r e w a s n ' t q u i t e s u r e t h e y e x i s t e d . A c t u a l l y , P o i n c a r e d e s c r i b e d h i s r e s e a r c h a s f o l l o w s . H e s a i d t h a t f o r f i f t e e n d a y s h e w o u l d w a k e u p i n t h e m o r n i n g a n d s i t a t h i s d e s k f o r a c o u p l e o f h o u r s t r y i n g t o c o n v i n c e h i m s e l f t h a t t h e a u t o m o r p h i c f o r m s h e i n v e n t e d c o u l d n ' t p o s s i b l y e x i s t . O n t h e f i f t e e n t h d a y , h e r e a l i z e d t h a t h e w a s w r o n g . T h e s e s t r a n g e f u n c t i o n s , h a r d t o i m a g i n e v i s u a l l y , d i d e x i s t . P o i n c a r e e x t e n d e d t h e m t o e v e n m o r e c o m p l i c a t e d f u n c t i o n s , c a l l e d m o d u l a r f o r m s . T h e m o d u l a r f o r m s l i v e o n t h e u p p e r h a l f o f t h e c o m p l e x p l a n e , a n d t h e y h a v e a h y p e r b o l i c g e o m e t r y . T h a t i s , t h e y l i v e i n a s t r a n g e s p a c e w h e r e t h e n o n -E u c l i d e a n g e o m e t r y o f B o l y a i a n d L o b a c h e v s k y r u l e s . T h r o u g h a n y p o i n t i n t h i s h a l f - p l a n e , v m a n y " l i n e s " p a r a l l e l t o a n y g i v e n l i n e e x i s t .

P o i n c a r e l e f t b e h i n d t h e s y m m e t r i c a u t o m o r p h i c f o r m s a n d t h e i r e v e n m o r e i n t r i c a t e m o d u l a r f o r m s a n d w e n t o n t o d o o t h e r m a t h e m a t i c s . H e w a s b u s y i n s o m a n y f i e l d s , o f t e n i n a f e w o f t h e m a t o n c e , t h a t h e h a d n o t i m e t o s i t a n d p o n d e r t h e b e a u t y o f h a r d l y i m a g i n a b l e , i n f i n i t e l y s y m m e t r i c e n t i t i e s . B u t u n b e k n o w n s t t o h i m , h e h a d j u s t s o w e d a n o t h e r s e e d i n t h e g a r d e n t h a t w o u l d e v e n t u a l l y b r i n g a b o u t t h e F e r m a t s o l u t i o n .

An Unexpected Connection with a Doughnut

I n 1922, t h e A m e r i c a n m a t h e m a t i c i a n L o u i s J.

M o r d e l l d i s c o v e r e d w h a t h e t h o u g h t w a s a v e r y s t r a n g e c o n n e c t i o n b e t w e e n

t h e s o l u t i o n s o f a l g e b r a i c e q u a t i o n s a n d t o p o l o g y . T h e e l e m e n t s o f t o p o l o g y a r e s u r f a c e s a n d s p a c e s . T h e s e s u r f a c e s c o u l d b e i n a n y d i m e n s i o n : t w o d i m e n s i o n s , s u c h a s t h e f i g u r e s i n a n c i e n t G r e e k g e o m e t r y , o r t h e y c o u l d b e i n t h r e e -d i m e n s i o n a l s p a c e , o r m o r e . T o p o l o g y i s a s t u d y o f c o n t i n u o u s f u n c t i o n s t h a t a c t o n t h e s e s p a c e s , a n d t h e p r o p e r t i e s o f t h e s p a c e s t h e m s e l v e s . T h e p a r t o f t o p o l o g y w h i c h c o n c e r n e d M o r d e l l w a s t h e o n e o f s u r f a c e s i n t h r e e -d i m e n s i o n a l s p a c e . A s i m p l e e x a m p l e o f s u c h a s u r f a c e i s a s p h e r e : i t i s t h e s u r f a c e o f a b a l l , s u c h a s a b a s k e t b a l l . T h e b a l l i s t h r e e -d i m e n s i o n a l , b u t i t s s u r f a c e ( a s s u m i n g n o d e p t h ) i s a t w o - d i m e n s i o n a l o b j e c t . T h e s u r f a c e o f t h e e a r t h i s a n o t h e r e x a m p l e . T h e e a r t h i t s e l f i s t h r e e d i m e n s i o n a l : a n y p l a c e o n t h e e a r t h o r i n s i d e i t c a n b e g i v e n b y i t s l o n g i t u d e ( o n e d i m e n s i o n ) , i t s l a t i t u d e ( a s e c o n d d i m e n s i o n ) , a n d i t s d e p t h ( t h e t h i r d d i m e n s i o n ) . B u t t h e s u r f a c e o f t h e e a r t h ( n o d e p t h ) i s t w o - d i m e n s i o n a l , s i n c e a n y p o i n t o n t h e s u r f a c e o f t h e e a r t h c a n b e s p e c i f i e d b y t w o n u m b e r s : i t s l o n g i t u d e a n d i t s l a t i t u d e .

T w o - d i m e n s i o n a l s u r f a c e s i n t h r e e - d i m e n s i o n a l s p a c e c a n b e c l a s s i f i e d a c c o r d i n g t o t h e i r g e n u s . T h e g e n u s i s

t h e n u m b e r o f h o l e s i n t h e s u r f a c e . T h e g e n u s o f a s p h e r e i s z e r o s i n c e t h e r e a r e n o h o l e s i n i t . A d o u g h n u t h a s o n e h o l e i n i t . T h e r e f o r e t h e g e n u s o f a d o u g h n u t ( m a t h e m a t i c a l l y c a l l e d a t o r u s ) i s o n e . A h o l e m e a n s a h o l e t h a t r u n s c o m p l e t e l y t h r o u g h t h e s u r f a c e . A c u p w i t h t w o h a n d l e s h a s t w o h o l e s t h r o u g h i t . T h e r e f o r e i t i s a s u r f a c e o f g e n u s t w o .

A s u r f a c e o f o n e g e n u s c a n b e t r a n s f o r m e d b y a c o n t i n u o u s f u n c t i o n i n t o a n o t h e r s u r f a c e o f t h e s a m e g e n u s . T h e o n l y w a y t o t r a n s f o r m a s u r f a c e o f o n e g e n u s t o o n e o f a d i f f e r e n t g e n u s

is by closing or opening some holes. This cannot be

done by a continuous function, since it will require

some ripping or fusing together, each of which is a

mathematical discontinuity.

Mordell discovered a strange and totally unexpected

connection between the number of holes in the

surface (the genus) of the space of solutions of an

equation and whether or not the

e q u a t i o n h a d a f i n i t e n u m b e r o r a n i n f i n i t e n u m b e r o f s o l u t i o n s . I f t h e s u r f a c e o f s o l u t i o n s , u s i n g c o m p l e x n u m b e r s f o r g r e a t e s t g e n e r a l i t y , h a d t w o o r m o r e h o l e s ( t h a t i s , h a d g e n u s t w o o r h i g h e r ) , t h e n t h e e q u a t i o n h a d o n l y f i n i t e l y m a n y w h o l e - n u m b e r s o l u t i o n s . M o r d e l l w a s u n a b l e t o p r o v e h i s d i s -c o v e r y , a n d i t b e c a m e k n o w n a s M o r d e l l ' s c o n j e c t u r e .

Faltings' Proof

I n 1983, a t w e n t y - s e v e n y e a r o l d G e r m a n m a t h e m a t i c i a n , G e r d F a l t i n g s , a t t h a t t i m e a t t h e U n i v e r s i t y o f W u p p e r t a l , w a s a b l e t o p r o v e t h e M o r d e l l c o n j e c t u r e . F a l t i n g s w a s n o t i n t e r e s t e d i n F e r m a t ' s L a s t T h e o r e m , c o n s i d e r i n g i t a n i s o l a t e d p r o b l e m i n n u m b e r t h e o r y . Y e t h i s p r o o f , w h i c h u s e d g r e a t i n g e n u i t y a l o n g w i t h t h e p o w e r f u l m a c h i n e r y o f a l g e b r a i c g e o m e t r y d e v e l o p e d i n t h i s c e n t u r y , h a d p r o f o u n d i m p l i c a t i o n s f o r t h e s t a t u s o f F e r m a t ' s L a s t T h e o r e m . B e c a u s e t h e g e n u s o f t h e F e r m a t e q u a t i o n f o r n g r e a t e r t h a n 3

w a s 2 o r m o r e , i t b e c a m e c l e a r t h a t i f i n t e g e r s o l u t i o n s t o t h e F e r m a t e q u a t i o n e x i s t e d a t a l l , t h e n t h e s e w e r e f i n i t e ( w h i c h w a s c o m f o r t i n g , s i n c e t h e i r n u m b e r w a s n o w l i m i t e d ) . S o o n a f t e r w a r d s , t w o m a t h e m a t i c i a n s , G r a n v i l l e a n d H e a t h - B r o w n , u s e d F a l t -

i n g s ' r e s u l t t o s h o w t h a t t h e n u m b e r o f s o l u t i o n s o f F e r m a t ' s e q u a t i o n , i f t h e y e x i s t e d , d e c r e a s e d a s t h e e x p o n e n t n i n c r e a s e d . I t w a s s h o w n t h a t t h e p r o p o r t i o n o f e x p o n e n t s f o r w h i c h F e r m a t ' s L a s t T h e o r e m w a s t r u e a p p r o a c h e d o n e h u n d r e d p e r c e n t a s n i n c r e a s e d .

I n o t h e r w o r d s , F e r m a t ' s L a s t T h e o r e m w a s " a l m o s t a l w a y s " t r u e . I f s o l u t i o n s t o F e r m a t ' s e q u a t i o n e x i s t e d ( i n w h i c h c a s e t h e T h e o r e m w a s n o t t r u e ) , t h e n s u c h s o l u t i o n s w e r e f e w a n d

v e r y f a r b e t w e e n . S o t h e s t a t u s o f F e r m a t ' s L a s t T h e o r e m i n 1983 w a s t h e f o l l o w i n g . T h e t h e o r e m w a s p r o v e n f o r n u p t o a m i l l i o n ( a n d i n 1992 t h e l i m i t w a s r a i s e d t o 4 m i l l i o n ) . I n a d d i -t i o n , f o r l a r g e r n, i f s o l u t i o n s e x i s t e d a t a l l t h e n t h e y w e r e v e r y f e w a n d d e c r e a s i n g w i t h n.

The Mysterious Greek General with the Funny Name T h e r e a r e d o z e n s o f e x c e l l e n t b o o k s o n m a t h e m a t i c s p u b -l i s h e d i n F r a n c e , w r i t t e n i n F r e n c h , w i t h t h e a u t h o r l i s t e d a s N i c o l a s B o u r b a k i . T h e r e w a s a G r e e k g e n e r a l n a m e d B o u r b a k i (1816-1897). I n 1862

B o u r b a k i w a s o f f e r e d t h e t h r o n e o f G r e e c e , w h i c h h e d e c l i n e d . T h e g e n e r a l h a d a n i m p o r t a n t r o l e i n t h e F r a n c o - P r u s s i a n w a r , a n d t h e r e i s a s t a t u e o f h i m i n t h e F r e n c h c i t y o f N a n c y . B u t G e n e r a l B o u r b a k i k n e w n o t h i n g a b o u t m a t h e m a t i c s . A n d h e n e v e r w r o t e a b o o k , a b o u t m a t h e m a t i c s o r a n y t h i n g e l s e . W h o w r o t e t h e m a n y v o l u m e s o f m a t h e m a t i c s b e a r i n g h i s n a m e ?

T h e a n s w e r l i e s i n t h e

h a p p y d a y s i n P a r i s o f b e t w e e n t h e t w o W o r l d W a r s . H e m i n g w a y , P i c a s s o a n d M a t i s s e w e r e n o t t h e o n l y p e o p l e w h o l i k e d t o s i t i n c a f e s a n d m e e t t h e i r f r i e n d s a n d s e e p e o p l e a n d b e s e e n . A t t h a t t i m e , a r o u n d t h e s a m e c a f e s o n t h e L e f t B a n k b y t h e U n i v e r s i t y o f P a r i s , t h e r e f l o u r i s h e d a v i b r a n t m a t h e m a t i c a l c o m m u n i t y . P r o f e s s o r s o f m a t h e m a t i c s f r o m t h e u n i v e r s i t y a l s o l i k e d t o m e e t t h e i r f r i e n d s , d r i n k a c a f e a u l a i t o r a p a s t i s i n a g o o d b r a s s e r i e o n t h e B o u l e v a r d S t . M i c h e l b y t h e b e a u t i f u l L u x e m b o u r g G a r d e n s , a n d d i s c u s s . . . m a t h e m a t i c s . S p r i n g t i m e i n P a r i s i n s p i r e d w r i t e r s , a r t i s t s , a n d m a t h e m a t i c i a n s . O n e i m a g i n e s t h a t o n a s u n n y d a y a t a p l e a s a n t c a f e a r o w d y g r o u p o f m a t h e m a t i c i a n s c o n g r e -

ss

g a t e d . F e e l i n g s o f f r a t e r n i t y o v e r c a m e t h e m a s t h e y a r g u e d a n i m a t e d l y o v e r s o m e f i n e p o i n t s o f a t h e o r y . T h e i r r e v e l r y p r o b a b l y d i s t u r b e d H e m i n g w a y w h o w r i t e s h e a l w a y s l i k e d t o w o r k b y h i m s e l f a t a c a f e , a n d h e p r o b a b l y l e f t t o g o t o o n e o f h i s a l t e r n a t e , l e s s f a v o r i t e h a u n t s . B u t t h e m a t h e m a t i c i a n s d i d n ' t c a r e . T h e y v a l u e d e a c h o t h e r ' s c o m p a n y , a n d a c a f e f u l l o f m a t h e m a t i c i a n s — a l l s p e a k i n g t h e s a m e l a n g u a g e o f n u m b e r s a n d s y m b o l s a n d s p a c e s a n d f u n c t i o n s — w a s i n t o x i c a t i n g . " T h i s i s w h a t t h e P y t h a g o r e a n s m u s t h a v e f e l t w h e n t h e y t a l k e d m a t h e m a t i c s , " p e r h a p s o n e o f t h e s e n i o r p e o p l e i n t h e g r o u p s a i d , a s h e l i f t e d h i s g l a s s i n a t o a s t . " Y e s , b u t t h e y d i d n ' t d r i n k P e r n o d , " s a i d a n o t h e r a n d e v e r y o n e l a u g h e d . " B u t w e c o u l d b e l i k e t h e m , " a n s w e r e d t h e f i r s t . " W h y d o n ' t w e f o r m o u r own s o c i e t y ? A s e c r e t o n e , n a t u r a l l y . " T h e r e w e r e v o i c e s o f a p p r o v a l a l l a r o u n d . S o m e o n e s u g g e s t e d t h e y b o r r o w t h e n a m e o f o l d G e n e r a l B o u r b a k i . T h e r e w a s a r e a s o n f o r t h i s s u g g e s t i o n . I n t h o s e d a y s , t h e m a t h e m a t i c s d e p a r t m e n t a t t h e U n i v e r s i t y d f P a r i s h a d a n a n n u a l t r a d i t i o n o f i n v i t i n g a p r o f e s s i o n a l a c t o r w h o p r e s e n t e d h i m s e l f t o t h e a s s e m b l e d f a c u l t y a n d g r a d u a t e s t u d e n t s a s N i c o l a s

B o u r b a k i . H e w o u l d t h e n d o a o n e - m a n s h o w c o n s i s t i n g o f a l o n g m o n o l o g u e o f m a t h e m a t i c a l d o u b l e - t a l k . S u c h p r e s e n t a t i o n s w e r e v e r y e n t e r t a i n i n g s i n c e t h e r i c h n e s s o f m o d e r n m a t h e m a t i c a l t h e o r y m a k e s f o r a v a s t v o c a b u l a r y t h a t i s b o t h d e s c r i p t i v e o f m a t h e m a t i c s a n d h a s d i f f e r e n t m e a n i n g s i n e v e r y d a y l i f e . O n e s u c h w o r d i s " d e n s e . " T h e r a t i o n a l n u m b e r s a r e s a i d t o b e d e n s e w i t h i n t h e r e a l n u m b e r s . T h i s m e a n s t h a t w i t h i n a n y n e i g h b o r h o o d o f r a t i o n a l n u m b e r s a r e i r r a t i o n a l n u m b e r s . B u t " d e n s e " m e a n s m a n y o t h e r t h i n g s i n e v e r y d a y l i f e .

Graduate students today like to play the same

games of double meaning, and they like to tell the

story of beautiful Poly Nomial who meets the smooth

operator Curly Pi (polynomial, smooth operator, and

curly pi are all mathematical terms).

And so the books these mathematicians wrote

together bore the name Nicolas Bourbaki. A Bourbaki

Seminar was initiated, where mathematical ideas and

theories were discussed not infrequently.

Membership in the society was supposed to be

anonymous, and mathematical results were to be

credited to the society in the name of Bourbaki,

rather than to individual members.

But the members of Bourbaki were not the

Pythagoreans. While the author of the textbooks was

Nicolas Bourbaki, research results such as theorems

and their proofs—which are far more prestigious than

books—were credited to the individual

mathematicians who achieved the results. One of the

first members of Bourbaki was Andre Weil (1906- ),

who later moved to the United States and the

Institute of Advanced Study at Princeton. His name

would never be too far away from the important

conjecture leading to the solution of the Fermat

problem.

Another of the founders of Bourbaki was the

French mathematician Jean Dieudonne, who like most

of the other "French only" members of the society

moved on to greener pastures at the universities in

the United States. Dieudonne, who was the principal

author of many of the books bearing the collective

name of Nicolas Bourbaki, epitomizes the clash

between the Bourbakites' quest for individual

anonymity and their individual egos. Dieudonne once

published a paper bearing the name

o f B o u r b a k i . T h e p a p e r w a s f o u n d t o c o n t a i n a n e r r o r , a n d s o D i e u d o n n e p u b l i s h e d a n o t e t i t l e d : " O n a n E r r o r o f N . B o u r b a k i , " a n d s i g n e d i t J . D i e u d o n n e .

T h e s c h i z o p h r e n i c n a t u r e o f t h e s o c i e t y — i t s m e m b e r s w e r e a l l F r e n c h , b u t m o s t o f t h e m l i v e d i n t h e U n i t e d S t a t e s — m a n i f e s t s i t s e l f i n M r . B o u r b a k i ' s a f f i l i a t i o n . U s u a l l y , p u b l i c a t i o n s o f N i c o l a s B o u r b a k i l i s t h i s a f f i l i a t i o n a s " t h e U n i v e r s i t y o f N a n c a g o , " a f u s i o n o f t h e n a m e o f t h e F r e n c h c i t y o f N a n c y w i t h t h a t o f C h i c a g o . B u t B o u r b a k i p u b l i s h e s o n l y i n F r e n c h , a n d w h e n i t s m e m b e r s m e e t , u s u a l l y a t a F r e n c h r e s o r t c i t y , t h e c o n v e r s a t i o n s a r e n o t o n l y i n F r e n c h , t h e y a r e i n t h e d i a l e c t o f t h e P a r i s i a n s t u d e n t s . T h e c h a u v i n i s m c a r r i e s i n t o t h e s e p a r a t e l i v e s o f t h e s e F r e n c h m a t h e -m a t i c i a n s l i v i n g i n A m e r i c a . A n d r e W e i l , a f o u n d i n g B o u r -b a k i t e , p u b l i s h e d m a n y i m p o r t a n t p a p e r s i n E n g l i s h . B u t h i s Collected Works, w h i c h h a d s o m e b e a r i n g o n t h e c o n j e c t u r e r e l a t e d t o t h e F e r m a t p r o b l e m , w a s p u b l i s h e d i n F r e n c h a n d t i t l e d Oeuvres'. W e i l ' s u n u s u a l a c t i o n s w o u l d h u r t o n e o f t h e p r i n c i p a l p l a y e r s i n o u r d r a m a , a n d W e i l w o u l d n o t r e c o v e r f r o m t h i s d e b a c l e .

T h e m e m b e r s o f B o u r b a k i m u s t b e g i v e n c r e d i t f o r t h e i r c o l l e c t i v e s e n s e o f h u m o r . S o m e f o r t y y e a r s a g o , t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y

r e c e i v e d a n a p p l i c a t i o n f o r i n d i v i d u a l m e m b e r s h i p f r o m M r . N i c o l a s B o u r b a k i . T h e s e c r e t a r y o f t h e S o c i e t y w a s i m p e r t u r b a b l e . H e r e p l i e d t h a t i f M r . B o u r b a k i w a n t e d t o j o i n t h e S o c i e t y h e w o u l d h a v e t o a p p l y a s a n i n s t i -t u t i o n a l m e m b e r ( w h i c h w a s m u c h m o r e e x p e n s i v e ) . B o u r b a k i d i d n o t w r i t e b a c k .

F E R M A T ' S L A S T T H E O R E M Elliptic

Curves

D i o p h a n t i n e p r o b l e m s — t h a t i s , p r o b l e m s r a i s e d b y e q u a t i o n s o f t h e f o r m g i v e n b y D i o p h a n t u s i n t h e t h i r d c e n t u r y — b e g a n t o b e s t u d i e d m o r e a n d m o r e i n t h e t w e n t i e t h c e n t u r y u s i n g m a t h e m a t i c a l e n t i t i e s c a l l e d e l l i p t i c c u r v e s . B u t e l l i p t i c c u r v e s h a v e n o t h i n g t o d o w i t h e l l i p s e s . T h e y w e r e o r i g i n a l l y u s e d c e n t u r i e s e a r l i e r i n c o n n e c t i o n w i t h e l l i p t i c f u n c t i o n s , w h i c h i n t u r n w e r e d e v i s e d t o h e l p c a l c u l a t e t h e p e r i m e t e r o f a n e l l i p s e . A s w i t h m a n y i n n o v a t i v e i d e a s i n m a t h e m a t i c s , t h e p i o n e e r i n t h i s f i e l d w a s n o n e o t h e r t h a n G a u s s .

O d d l y , e l l i p t i c c u r v e s a r e n e i t h e r e l l i p s e s n o r e l l i p t i c f u n c t i o n s — t h e y a r e c u b i c p o l y n o m i a l s i n t w o v a r i a b l e s . T h e y l o o k l i k e : y 2

= a x 3 + b x 2 + c x w h e r e a, b, a n d c a r e i n t e g e r s o r r a t i o n a l n u m b e r s ( w e a r e c o n c e r n e d w i t h e l l i p t i c c u r v e s o v e r t h e r a t i o n a l n u m b e r s ) . E x a m p l e s o f s u c h e l l i p t i c c u r v e s a r e s h o w n b e l o w . 1 3

W h e n o n e l o o k s a t t h e r a t i o n a l p o i n t s o n t h e e l l i p t i c c u r v e s — t h a t i s , o n e l o o k s o n l y a t p o i n t s o n t h e c u r v e t h a t a r e r a t i o s o f t w o i n t e g e r s ( n o i r r a t i o n a l n u m b e r s s u c h a s p i o r t h e s q u a r e r o o t o f t w o , e t c . ) , t h e s e n u m b e r s f o r m a g r o u p . T h a t m e a n s t h a t t h e y h a v e n i c e p r o p e r t i e s . T a k e a n y t w o s o l u t i o n s , - t h e y c a n b e " a d d e d " i n a s e n s e t o p r o d u c e a third s o l u t i o n o n t h e c u r v e . N u m b e r t h e o r i s t s h a v e b e c o m e f a s c i n a t e d w i t h t h e e l l i p t i c c u r v e s s i n c e

t h e y c a n a n s w e r m a n y q u e s t i o n s a b o u t e q u a t i o n s a n d t h e i r s o l u t i o n s . E l l i p t i c c u r v e s t h u s b e c a m e o n e o f t h e f o r e m o s t r e s e a r c h t o o l s i n n u m b e r t h e o r y . 1 4

A Strange Conjecture Is about to be Made

I t w a s k n o w n f o r s o m e t i m e b y n u m b e r t h e o r y e x p e r t s t h a t some o f t h e e l l i p t i c c u r v e s t h e y w e r e s t u d y i n g w e r e m o d u l a r .

T h a t i s , t h e s e f e w e l l i p t i c c u r v e s c o u l d b e v i e w e d a s c o n n e c t e d w i t h m o d u l a r f o r m s . S o m e e l l i p t i c c u r v e s c o u l d b e s o m e h o w c o n n e c t e d w i t h t h e c o m p l e x p l a n e a n d t h e s e f u n c t i o n s i n h y p e r b o l i c s p a c e w i t h t h e i r m a n y s y m m e t r i e s . I t w a s n o t c l e a r w h y a n d h o w t h i s w a s h a p p e n i n g . T h e m a t h e m a t i c s w a s e x t r e m e l y c o m p l i c a t e d , e v e n f o r e x p e r t s , a n d t h e i n t e r n a l s t r u c t u r e — t h e b e a u t i f u l h a r m o n i e s t h a t e x i s t e d — w a s l i t t l e u n d e r s t o o d . T h e e l l i p t i c f u n c t i o n s t h a t w e r e i n d e e d m o d u l a r h a d i n t e r e s t i n g p r o p e r t i e s . S o o n s o m e o n e w o u l d m a k e t h e b o l d c o n j e c t u r e t h a t all e l l i p t i c f u n c t i o n s w e r e m o d u l a r .

T o u n d e r s t a n d t h e i d e a o f m o d u l a r i t y , w h i c h e x i s t s

w i t h i n

t h e n o n - E u c l i d e a n s p a c e o f t h e u p p e r c o m p l e x h a l f -p l a n e , w h e r e s y m m e t r i e s a r e s o c o m p l i c a t e d t h a t t h e y c a n h a r d l y b e v i s u a l i z e d , i t i s u s e f u l t o l o o k a t a v e r y s i m p l e e x a m p l e . T h i s i s a n e x a m p l e w h e r e t h e c u r v e o f i n t e r e s t i s n o t a n e l l i p t i c c u r v e , - i n s t e a d o f a c u b i c e q u a t i o n i n t w o v a r i a b l e s , i t i s o n l y a s q u a r e d e q u a t i o n i n t w o v a r i a b l e s : t h e c u r v e i s a s i m p l e c i r c l e . T h e e q u a t i o n o f a c i r c l e w i t h r a d i u s a2

w h o s e c e n t e r i s z e r o i s g i v e n b y : x 2 + y2 =a2. N o w l o o k a t t h e s i m p l e p e r i o d i c f u n c t i o n s : x =a c o s ( , a n d y = a s i n t . T h e s e t w o f u n c t i o n s c a n s t a n d f o r x a n d y i n t h e e q u a t i o n o f t h e c i r c l e . T h e e q u a t i o n o f t h e c i r c l e i s m o d u l a r i n t h i s s e n s e . T h e r e a s o n i s t h e t r i g o n o m e t r i c i d e n t i t y t h a t s a y s t h a t s i n 2 t + c o s 2 t = 1, a n d s u b s t i t u t i n g t h i s f o r -m u l a i n t o t h e e q u a t i o n o f t h e c i r c l e ( e a c h t e r m m u l t i p l i e d b y a) g i v e s a n i d e n t i t y .

A m o d u l a r e l l i p t i c c u r v e i s j u s t a n e x t e n s i o n o f t h i s i d e a t o t h e m o r e c o m p l i c a t e d c o m p l e x p l a n e , w i t h a s p e c i a l n o n - E u c l i d e a n g e o m e t r y . H e r e t h e p e r i o d i c f u n c t i o n s a r e s y m m e t r i e s n o t o n l y w i t h r e s p e c t t o o n e v a r i a b l e , t , a s w i t h t h e s i n e s a n d c o s i n e s o n t h e l i n e— t h e y a r e t h e a u t o m o r p h i c , o r t h e m o d u l a r f o r m s o n t h e c o m p l e x p l a n e , w h i c h h a v e s y m m e t r i e s w i t h r e s p e c t t o m o r e c o m p l i c a t e d

t r a n s f o r m a t i o n s ( f ( z ) —>j(az+b/cz+d)).

Tokyo, Japan, Early 1950s

I n t h e e a r l y 1950s, J a p a n w a s a n a t i o n e m e r g i n g f r o m t h e d e v a s t a t i o n o f w a r . P e o p l e w e r e n o l o n g e r h u n g r y , b u t t h e y w e r e s t i l l p o o r a n d e v e r y d a y s u r v i v a l w a s a s t r u g g l e f o r t h e a v e r a g e J a p a n e s e . Y e t f a c t o r i e s w e r e b e i n g r e b u i l t f r o m t h e r u b b l e , b u s i n e s s e s r e e s t a b l i s h e d , a n d t h e g e n e r a l m o o d w a s h o p e f u l . U n i v e r s i t y l i f e i n J a p a n a t t h a t t i m e w a s a l s o d i f f i c u l t . C o m

p e t i t i o n a m o n g s t u d e n t s w a s f i e r c e : g o o d g r a d e s m e a n t g o o d j o b s a f t e r g r a d u a t i o n . T h i s w a s e s p e c i a l l y t r u e f o r d o c t o r a l s t u d e n t s i n p u r e m a t h e m a t i c s , s i n c e p o s i t i o n s a t u n i v e r s i t i e s w e r e s c a r c e e v e n t h o u g h t h e p a y w a s l o w . Y u t a k a T a n i y a m a w a s o n e s u c h d o c t o r a l s t u d e n t i n m a t h e m a t i c s . H e w a s b o r n o n N o v e m b e r 12, 1927, t h e y o u n g e s t o f e i g h t c h i l d r e n i n t h e f a m i l y o f a c o u n t r y d o c t o r i n t h e t o w n o f K i s a i , a b o u t 3 0 m i l e s n o r t h o f T o k y o . A t a y o u n g a g e , T a n i y a m a b e g a n t o s t u d y t h e a r e a o f m a t h e m a t i c s i n v o l v i n g c o m p l e x m u l t i p l i c a t i o n o f a b e l i a n v a r i e t i e s . N o t m u c h w a s k n o w n a b o u t t h i s f i e l d a n d T a n i y a m a h a d a v e r y d i f f i c u l t t i m e . T o m a k e t h i n g s w o r s e , h e f o u n d t h e a d v i c e o f o l d e r p r o f e s s o r s a t t h e U n i v e r s i t y o f T o k y o v i r t u a l l y u s e l e s s . H e h a d t o d e r i v e e v e r y d e t a i l o n h i s o w n a n d h e u s e d t o d e s c r i b e e v e r y t a s k i n h i s m a t h e m a t i c a l r e s e a r c h u s i n g f o u r C h i n e s e c h a r a c t e r s t h a t m e a n t " h a r d f i g h t i n g " a n d " b i t t e r s t r u g g l e . " N o t h i n g w a s e a s y i n y o u n g Y u t a k a T a n i y a m a ' s l i f e .

T a n i y a m a l i v e d i n a o n e -r o o m a p a r t m e n t o f 81 s q u a r e f e e t . T h e r e w a s o n l y o n e t o i l e t o n e v e r y f l o o r o f t h e b u i l d i n g , s h a r e d b y a l l t h e r e s i d e n t s o f t h e f l o o r . T o t a k e a b a t h , T a n i y a m a h a d t o g o t o a p u b l i c b a t h h o u s e

s o m e d i s t a n c e a w a y f r o m h i s b u i l d i n g . T h e s h a b b y a p a r t m e n t b u i l d i n g w a s n a m e d " V i l l a T r a n q u i l M o u n t a i n s , " i r o n i c a l l y s o s i n c e i t s t o o d o n a b u s y s t r e e t a n d b y a r a i l r o a d t r a c k o n w h i c h t r a i n s t h u n d e r e d b y e v e r y f e w m i n u t e s . P o s s i b l y s o h e c o u l d c o n c e n t r a t e b e t t e r o n h i s r e s e a r c h , y o u n g Y u t a k a w o r k e d m o s t l y a t n i g h t , o f t e n g o i n g t o b e d a t 6 AM

w h e n t h e n o i s y d a y b e g a n . E x c e p t i n t h e h e a t o f s u m m e r , a l m o s t e v e r y d a y T a n i y a m a w o r e t h e s a m e b l u e - g r e e n s u i t w i t h a m e t a l l i c s h e e n . A s h e e x p l a i n e d t o h i s g o o d f r i e n d G o r o S h i m u r a , h i s

f a t h e r b o u g h t t h e m a t e r i a l v e r y c h e a p l y f r o m a p e d d l e r . B u t b e c a u s e o f t h e l u s t e r , n o o n e i n t h e f a m i l y d a r e d w e a r i t . Y u t a k a , w h o d i d n ' t c a r e h o w h e l o o k e d , f i n a l l y v o l u n t e e r e d a n d h a d t h e m a t e r i a l t a i l o r e d f o r a s u i t w h i c h b e c a m e h i s d a i l y o u t f i t .

T a n i y a m a g r a d u a t e d f r o m t h e U n i v e r s i t y o f T o k y o i n 1953 a n d t o o k o n t h e p o s i t i o n o f " s p e c i a l r e s e a r c h s t u d e n t " a t t h e d e p a r t m e n t o f m a t h e m a t i c s t h e r e . H i s f r i e n d S h i m u r a g r a d u a t e d a y e a r e a r l i e r a n d h a d a s i m i l a r p o s i t i o n i n m a t h e m a t i c s a t t h e C o l l e g e o f G e n e r a l E d u c a t i o n a c r o s s c a m p u s . T h e i r f r i e n d s h i p b e g a n a f t e r o n e o f t h e m w r o t e t h e o t h e r a l e t t e r a s k i n g h i m t o r e t u r n t o t h e l i b r a r y a n i s s u e o f a m a t h e m a t i c a l j o u r n a l w h i c h i n t e r e s t e d t h e m b o t h . T h e y w o u l d o f t e n e a t t o g e t h e r a t i n e x p e n s i v e r e s t a u r a n t s s u p p o s e d l y s e r v i n g W e s t e r n -s t y l e f o o d , s u c h a s t o n g u e s t e w , w h i c h w a s b e c o m i n g p o p u l a r i n J a p a n . 1 5

F e w g o o d m a t h e m a t i c i a n s s t a y e d i n J a p a n i n t h o s e d a y s . O n c e a m a t h e m a t i c i a n a c h i e v e d s o m e r e n o w n , h e o r s h e w o u l d t r y t o m o v e t o a u n i v e r s i t y i n A m e r i c a o r E u r o p e , w h e r e t h e m a t h e m a t i c a l c o m m u n i t y w a s m o r e e s t a b l i s h e d a n d w h e r e c o n n e c t i o n s w i t h p e o p l e d o i n g r e s e a r c h i n t h e s a m e f i e l d s w e r e p o s s i b l e . S u c h l i n k s w e r e i m p o r t a n t f o r

c o n d u c t i n g r e s e a r c h i n e s o t e r i c a r e a s a b o u t w h i c h n o t m u c h w a s k n o w n . T o t r y t o f o s t e r r e s e a r c h t i e s w i t h p e o p l e w i t h k n o w l e d g e i n t h e i r f i e l d o f i n t e r e s t , t h e t w o f r i e n d s h e l p e d o r g a n i z e t h e T o k y o - N i k k o S y m p o s i u m o n A l g e b r a i c N u m b e r T h e o r y i n S e p t e m b e r , 1955. S o m e s t a t e m e n t s m a d e a t t h i s s m a l l c o n f e r e n c e , w h i l e d e s t i n e d t o r e m a i n o b s c u r e f o r a l o n g t i m e , w o u l d e v e n t u a l l y l e a d t o m o m e n t o u s r e s u l t s — a n d a s a v a g e c o n t r o v e r s y — a l m o s t f o r t y y e a r s l a t e r .

Goro Shimura, circa 1965, when he first

developed his conjecture

A Hopeful Beginning

T h e t w o f r i e n d s h e l p e d f i l e t h e n e c e s s a r y f o r m s w i t h t h e a d m i n i s t r a t i o n , a r r a n g e d f o r t h e c o n f e r e n c e f a c i l i t i e s , a n d h e l p e d s e n d o u t i n v i t a t i o n s t o l o c a l a n d f o r e i g n m a t h e m a t i c i a n s t h e y h o p e d w o u l d a t t e n d t h e c o n f e r e n c e . A n d r e W e i l , w h o h a d l e f t F r a n c e i n t h e m e a n t i m e a n d w a s a p r o f e s s o r a t t h e U n i v e r s i t y o f C h i c a g o , w a s o n e o f t h o s e i n v i t e d t o a t t e n d . A t t h e I n t e r n a -t i o n a l C o n g r e s s o f M a t h e m a t i c i a n s f i v e y e a r s e a r l i e r , W e i l h a d b r o u g h t t o

t h e a t t e n t i o n o f t h e m a t h e m a t i c a l c o m m u n i t y a n

u n k n o w n c o n j e c t u r e b y a m a t h e m a t i c i a n b y t h e n a m e o f H a s s e a b o u t t h e " z e t a f u n c t i o n o f a v a r i e t y o v e r a n u m b e r f i e l d . " T h e o b s c u r e s t a t e m e n t h e l d s o m e i n t e r e s t t o r e s e a r c h e r s i n n u m b e r t h e o r y . A p p a r e n t l y , W e i l w a s c o l l e c t i n g t h e s e c o n j e c t u r a l i d e a s i n t h e t h e o r y o f n u m b e r s a n d i n c l u d e d t h i s o n e i n h i s Collected Papers, g i v i n g c r e d i t t o H a s s e .

H i s i n t e r e s t i n r e s u l t s i n t h i s a r e a m a d e h i m a t t r a c t i v e t o T a n i y a m a a n d S h i m u r a , a n d t h e y w e r e p l e a s e d w h e n h e a c c e p t e d t h e i n v i t a t i o n t o a t t e n d t h e i r c o n f e r e n c e . A n o t h e r f o r e i g n m a t h e m a t i c i a n t o c o m e t o T o k y o - N i k k o w a s a y o u n g e r F r e n c h m a t h e m a t i c i a n , J e a n - P i e r r e S e r r e . W h i l e h e m a y n o t h a v e b e e n a m e m b e r o f B o u r b a k i a t t h a t t i m e , s i n c e t h e s o c i e t y i n c l u d e d o n l y w e l l -k n o w n m a t h e m a t i c i a n s , h e w o u l d b e c o m e o n e w i t h i n t h e f o l l o w i n g d e c a d e s . S e r r e h a s b e e n d e s c r i b e d b y s o m e m a t h e m a t i c i a n s a s a m b i t i o u s a n d f i e r c e l y c o m p e t i t i v e . H e c a m e t o T o k y o - N i k k o t o l e a r n a s m u c h a s h e c o u l d . T h e J a p a n e s e k n e w s o m e t h i n g s a b o u t n u m b e r t h e o r y , a n d t h e y h a d m a n y r e s u l t s p u b l i s h e d o n l y i n J a p a n e s e a n d t h u s h i d d e n f r o m t h e r e s t o f t h e w o r l d . T h i s w a s a g r e a t o p p o r t u n i t y t o l e a r n o f t h e s e r e s u l t s , s i n c e t h e c o n f e r e n c e w a s t o b e

c o n d u c t e d i n E n g l i s h . H e w o u l d b e o n e o f f e w p e o p l e o u t s i d e J a p a n w i t h k n o w l -e d g e o f t h e m a t h e m a t i c s p r e s e n t e d t h e r e . W h e n t h e p r o c e e d i n g s o f t h e s y m p o s i u m w o u l d b e p u b l i s h e d , i t w o u l d b e i n J a p a n e s e o n l y . T w e n t y y e a r s l a t e r , S e r r e w o u l d d r a w a t t e n t i o n t o s o m e e v e n t s a t T o k y o - N i k k o , a n d t h e w o r l d w o u l d h e a r h i s v e r s i o n — n o t t h e o n e r e c o r d e d i n t h e J a p a n e s e p r o c e e d i n g s .

T h e p r o c e e d i n g s i n c l u d e d t h i r t y - s i x p r o b l e m s . P r o b l e m s 10, 11, 12, a n d 13 w e r e w r i t t e n b y Y u t a k a T a n i y a m a . S i m i l a r t o t h e i d e a s o f H a s s e , T a n i y a m a ' s p r o b l e m s c o n s t i t u t e d a c o n j e c t u r e

a b o u t z e t a f u n c t i o n s . H e s e e m e d t o c o n n e c t P o i n c a r e ' s a u t o m o r p h i c f u n c t i o n s o f t h e c o m p l e x p l a n e w i t h t h e z e t a f u n c t i o n o f a n e l l i p t i c c u r v e . I t w a s m y s t e r i o u s t h a t a n e l l i p t i c c u r v e s h o u l d s o m e h o w b e c o n n e c t e d w i t h s o m e t h i n g i n t h e c o m p l e x p l a n e .

"You Are Saying What...?"

T h e c o n j e c t u r e e m b o d i e d i n t h e f o u r p r o b l e m s w a s n e b u l o u s . T a n i y a m a d i d n o t f o r m u l a t e t h e p r o b l e m s i n a v e r y m e a n i n g f u l w a y , p o s s i b l y b e c a u s e h e w a s n ' t q u i t e s u r e w h a t t h e c o n n e c -t i o n w a s . B u t t h e b a s i c i d e a w a s t h e r e . I t w a s a n i n t u i t i o n , a g u t f e e l i n g t h a t t h e a u t o m o r p h i c f u n c t i o n s w i t h t h e i r m a n y s y m m e t r i e s o n t h e c o m p l e x p l a n e w e r e s o m e h o w c o n n e c t e d w i t h t h e e q u a t i o n s o f D i o p h a n t u s . I t c e r t a i n l y w a s n ' t o b v i o u s . H e w a s p o s i t i n g a h i d d e n c o n n e c t i o n b e t w e e n t w o v e r y d i f f e r e n t b r a n c h e s o f m a t h e m a t i c s .

A n d r e W e i l w a n t e d t o k n o w e x a c t l y w h a t T a n i y a m a h a d i n m i n d . A c c o r d i n g t o t h e w r i t t e n r e c o r d o f t h e c o n f e r e n c e , t h e p r o c e e d i n g s p u b l i s h e d i n J a p a n e s e , t h e f o l l o w i n g e x c h a n g e t o o k p l a c e b e t w e e n W e i l a n d T a n i y a m a : 1 6

Weil: Do you think all elliptic functions are

uniformized by modular functions?

Taniyama: Modular functions alone will not be

enough. I think other special types of automorphic

functions are necessary. Weil: Of course some of

them can probably be handled that way. But in the

general case, they look completely different and

mysterious...

T w o t h i n g s a r e e v i d e n t f r o m t h e c o n v e r s a t i o n . F i r s t , T a n i y a m a

w a s r e f e r r i n g t o " a u t o m o r p h i c f u n c t i o n s " r a t h e r t h a n " m o d u l a r f u n c t i o n s a l o n e " a s b e i n g a s s o c i a t e d w i t h t h e e l l i p t i c c u r v e s . A n d s e c o n d , W e i l d i d n o t b e l i e v e t h a t i n g e n e r a l t h e r e w a s s u c h a c o n n e c t i o n . L a t e r h e w o u l d b e m o r e s p e c i f i c a b o u t t h i s d i s b e l i e f , a l l o f w h i c h w o u l d m a k e i t a s t o u n d i n g t h a t his n a m e , o f a l l p e o p l e ' s , s h o u l d e n d u p b e i n g a s s o c i a t e d w i t h a c o n j e c t u r e h e n e i t h e r f o r m u l a t e d n o r e v e n b e l i e v e d w a s t r u e . B u t f a t e s o m e t i m e s t a k e s s t r a n g e , i m p l a u s i b l e t u r n s , a n d e v e n m o r e b i z a r r e o c c u r r e n c e s w e r e g o i n g t o t r a n s p i r e .

A l l t h i s w o u l d m a t t e r d e c a d e s l a t e r . T o k n o w e x a c t l y w h a t Y u t a k a T a n i y a m a m e a n t , t h o u g h t , a n d s a i d w o u l d b e a b o o n t o m o d e r n h i s t o r i a n s . B u t , u n f o r t u n a t e l y , t r a g e d y s t a l k e d T a n i y a m a , a s i t d i d s o m a n y o t h e r y o u n g m a t h e m a t i c a l g e n i u s e s .

W i t h i n a c o u p l e o f y e a r s , G o r o S h i m u r a l e f t T o k y o , f i r s t f o r P a r i s , t h e n f o r t h e I n s t i t u t e f o r A d v a n c e d S t u d y a n d P r i n c e t o n U n i v e r s i t y . T h e t w o f r i e n d s c o n t i n u e d t o c o m m u n i c a t e b y m a i l . I n S e p t e m b e r , 1958, G o r o S h i m u r a r e c e i v e d t h e l a s t l e t t e r w r i t t e n b y Y u t a k a T a n i y a m a . I n t h e m o r n i n g o f N o v e m b e r 17, 1958, f i v e d a y s a f t e r h i s t h i r t y - f i r s t b i r t h d a y , Y u t a k a T a n i y a m a w a s f o u n d d e a d i n

h i s a p a r t m e n t , a s u i c i d e n o t e o n h i s d e s k .

Shimura's Conjecture

A d e c a d e p a s s e d s i n c e t h e T o k y o - N i k k o c o n f e r e n c e a n d G o r o S h i m u r a , n o w a t P r i n c e t o n , c o n t i n u e d h i s r e s e a r c h o n n u m b e r t h e o r y , z e t a f u n c t i o n s , a n d e l l i p t i c c u r v e s . H e u n d e r s t o o d w h e r e h i s l a t e f r i e n d h a d b e e n w r o n g , a n d h i s o w n r e s e a r c h a n d q u e s t f o r h i d d e n h a r m o n i e s i n t h e r e a l m s o f m a t h e m a t i c s l e d h i m t o f o r m u l a t e a d i f f e r e n t , b o l d e r a n d m o r e p r e c i s e c o n j e c t u r e . H i s c o n j e c t u r e w a s t h a t e v e r y e l l i p t i c c u r v e o v e r t h e

r a t i o n a l n u m b e r s i s u n i f o r m i z e d b y a m o d u l a r f o r m . M o d u l a r f o r m s a r e m o r e s p e c i f i c e l e m e n t s o v e r t h e c o m p l e x p l a n e t h a n a r e t h e a u t o m o r p h i c f u n c t i o n s o f T a n i y a m a . A n d s p e c i f y i n g t h e d o m a i n a s t h e r a t i o n a l n u m b e r s , a n d o t h e r m o d i f i c a t i o n s , w e r e i m p o r t a n t c o r r e c t i o n s a s w e l l .S h i m u r a ' s c o n j e c t u r e c a n b e

e x p l a i n e d u s i n g a p i c t u r e :

I f w e " f o l d " t h e c o m p l e x p l a n e a s a t o r u s ( t h e d o u g h n u t i n t h e p i c t u r e ^ , t h e n t h i s s u r f a c e w i l l h o l d a l l s o l u t i o n s t o e l l i p t i c e q u a t i o n s o v e r t h e r a t i o n a l n u m b e r s , t h e s e i n t u r n a r i s i n g f r o m t h e e q u a t i o n s o f D i o p h a n t u s . W h a t w o u l d l a t e r b e i m p o r t a n t t o t h e s o l u t i o n o f F e r m a t ' s L a s t T h e o r e m i s t h a t i f a s o l u t i o n t o F e r m a t ' s e q u a t i o n x n + y n

= z n e x i s t e d , t h i s s o l u t i o n w o u l d a l s o h a v e t o l i e o n t h a t t o r u s . N o w , S h i m u r a c o n j e c t u r e d t h a t e v e r y e l l i p t i c c u r v e w i t h r a t i o n a l c o e f f i c i e n t s ( t h a t i s , a n e q u a t i o n w i t h c o e f f i c i e n t s o f

t h e f o r m alb w h e r e b o t h a a n d b

a r e i n t e g e r s ) h a s a " m a t e " o n t h e c o m p l e x h a l f - p l a n e o f P o i n c a r e , w i t h i t s n o n -E u c l i d e a n , h y p e r b o l i c g e o m e t r y . T h e p a r t i c u l a r m a t e o f e a c h r a t i o n a l e l l i p t i c c u r v e w a s a v e r y s p e c i f i c f u n c t i o n o n t h e

c o m p l e x h a l f - p l a n e , w h i c h w a s i n v a r i a n t u n d e r c o m p l i c a t e d t r a n s f o r m a t i o n s o f t h e p l a n e — t h e o n e s m e n t i o n e d e a r l i e r :

/ ( z ) >f{az+b/cz+d), t h e c o e f f i c i e n t s f o r m i n g a g r o u p

w i t h m a n y u n e x p e c t e d s y m m e t r i e s . A l l o f t h i s w a s v e r y c o m p l e x , v e r y t e c h n i c a l , a n d — a s m o s t m a t h e m a t i c i a n s w o u l d b e l i e v e f o r s e v e r a l d e c a d e s —i m p o s s i b l e t o p r o v e i n t h e f o r e s e e a b l e f u t u r e .W h a t S h i m u r a ' s c o n j e c t u r e

w a s s a y i n g w a s t h a t e v e r y e l l i p t i c c u r v e w a s t h e p a r t o f a n i c e b e r g l y i n g a b o v e t h e w a t e r l i n e . B e l o w t h e s u r f a c e l a y a w h o l e i n t r i c a t e s t r u c t u r e . T o p r o v e t h e c o n j e c t u r e , o n e w o u l d h a v e t o s h o w t h a t every i c e b e r g h a d a n u n d e r w a t e r p a r t . S o m e s p e c i a l g r o u p s o f i c e b e r g s w e r e k n o w n t o h a v e t h e u n d e r w a t e r p a r t , b u t s i n c e t h e r e w e r e i n f i n i t e l y m a n y i c e b e r g s , o n e c o u l d n ' t j u s t g o l o o k u n d e r e a c h o n e o f t h e m . A g e n e r a l p r o o f w a s n e c e s s a r y t o s h o w t h a t a n i c e b e r g c o u l d n ' t e x i s t w i t h o u t p a r t o f i t b e i n g u n d e r w a t e r . T h e f o r m u l a t i o n o f s u c h a p r o o f w a s c o n s i d e r e d e x c e e d i n g l y d i f f i c u l t .

Intrigue and a Betrayal

A t a p a r t y a t t h e I n s t i t u t e f o r A d v a n c e d S t u d y a t P r i n c e t o n i n t h e e a r l y 1960s,

S h i m u r a a g a i n m e t J e a n -P i e r r e S e r r e . A c c o r d i n g t o

S h i m u r a , S e r r e a p p r o a c h e d h i m r a t h e r a r r o g a n t l y . " I d o n ' t t h i n k t h a t y o u r r e s u l t s o n m o d u l a r c u r v e s a r e a n y g o o d , " h e s a i d . " W h y , t h e y d o n ' t e v e n a p p l y t o a n a r b i t r a r y e l l i p t i c c u r v e . " I n r e s p o n s e , S h i m u r a s t a t e d h i s c o n j e c t u r e e x a c t l y : " S u c h a c u r v e s h o u l d always b e u n i f o r m i z e d b y a m o d u l a r c u r v e . " 1 7 S e r r e w e n t t o W e i l , w h o w a s n o t a t t h e p a r t y b u t w a s a m e m b e r o f t h e I n s t i -t u t e a n d t h e r e f o r e i n t h e i m m e d i a t e a r e a , a n d t o l d h i m o f h i s c o n v e r s a t i o n w i t h S h i m u r a . I n r e s p o n s e , A n d r e W e i l c a m e t o

Shimura. "Did you really say that?" he asked him, puzzled. "Yes,"

answered Shimura, "don't you think it plausible?" Ten years after

Taniyama's related conjecture, Andre Weil still did not believe

either conjecture. He answered: "I don't see any reason against it,

since one and the other of these sets are denumerable, but I don't

see any reason either for this hypothesis." What Weil said on this

occasion would later be described as "stupid," and "inane" by Serge

Lang of Yale University, who would circulate these comments

together with copies of two dozen letters he named collectively

"The Taniyama-Shimura File," among about fifty mathematicians

worldwide. What Weil meant in his response to Shimura was

tantamount to the following: If in a room you have seven men and

seven women and you conjecture that these are seven married

couples, then 1 see no reason against it, since the number of men

is the same as the number of women. But I don't see any reason for

your hypothesis, either. It could be that they are all single. What

made Lang describe the statement as stupid was that the counting

argument didn't really apply in any simple way here, because

"denumerable" means infinite and countable (such as the number

of all the positive integers: 1, 2, 3, 4,...) and matching such

infinite collections is no simple task. At any rate, it is clear that

Andre Weil did not believe Shimura's theory was necessarily true.

He would later admit that the conversation took place and, stupid,

inane, or otherwise, would quote it. But this would happen only in

1979, when he would write:18

Quelques annees plus tard, a Princeton, Shimura me demanda si

je trouvais plausible que toute courbe elliptique sur Q hit

contenue dans le jacobienne d'une courbe definie par une sous-

groupe de

congruence du groupe modulaire, je lui repondis, il

me semble, que je n'y voyais pas d'empechement,

puisque l'un et l'autre ensemble est denombrable,

mais je ne voyais rien non plus qui parlat en faveur

de cette hypothese.

[ " S o m e y e a r s l a t e r , a t P r i n c e t o n , S h i m u r a a s k e d m e i f I f o u n d i t p l a u s i b l e t h a t e v e r y e l l i p t i c c u r v e o v e r Q w a s c o n t a i n e d i n t h e j a c o b i a n o f a c u r v e d e f i n e d b y a c o n g r u e n c e s u b g r o u p o f a m o d u l a r g r o u p , - 1 r e s p o n d e d t o h i m , i t s e e m s t o m e , t h a t I d o n ' t s e e a n y t h i n g a g a i n s t i t , s i n c e o n e s e t a n d t h e o t h e r a r e d e n u m e r a b l e , b u t n e i t h e r d o 1 s e e a n y t h i n g t h a t s p e a k s i n f a v o r o f t h i s h y p o t h e s i s . " ]

B u t e v e n t h e n , W e i l w o u l d w r i t e " S h i m u r a a s k e d m e " ( m e demanda), r a t h e r t h a n " S h i m u r a t o l d m e , " w h e n r e f e r r i n g t o t h e s t a t e m e n t t h a t i s S h i m u r a ' s c o n j e c t u r e . W e i l p u b l i s h e d s o m e r e l a t e d p a p e r s , a n d w h i l e h e d i d n o t b e l i e v e S h i m u r a ' s t h e o r y , h i s o w n n a m e b e c a m e a s s o c i a t e d w i t h i t . T h e e r r o r w a s p e r p e t u a t e d w h e n m a t h e m a t i c i a n s m a d e r e f e r e n c e s i n t h e i r p a p e r s t o t h e w o r k s o f o t h e r s , a n d t h e m i s q u o t a t i o n i s p r e s e n t t o t h i s d a y w h e n w r i t e r s i g n o r a n t o f t h e h i s t o r y r e f e r t o t h e W e i l - T a n i y a m a c o n j e c t u r e i n s t e a d o f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e . W e i l s e e m e d t o e n j o y h i s a s s o c i a t i o n w i t h a n

i m p o r t a n t t h e o r y w h i c h —w h i l e h e h i m s e l f d i d n o t b e l i e v e i n i t — m o s t m a t h e m a t i c i a n s t h o u g h t w o u l d b e p r o v e d s o m e d a y i n t h e d i s t a n t f u t u r e .

W i t h t h e p a s s i n g d e c a d e s , t h e r e w a s m o r e a n d m o r e r e a s o n f o r t h e c o n n e c t i o n t o e x i s t . I f a n d w h e n t h e c o n j e c t u r e w a s p r o v e d , i t w o u l d b e a s u b s t a n t i a l m a t h e m a t i c a l t h e o r y . W e i l w o r k e d a r o u n d t h e c o n j e c t u r e , n e v e r l e a v i n g m a t h e m a t i c a l

results he obtained too far away from the possible

connection between modular forms of the complex

plane and the elliptic curves of Diophantine

equations. And while he certainly knew better, he

held back references to Shimura and his crucial role

until almost two decades had passed. Then he gave

offhand praise to Shimura in casual conversation and

mentioned him— almost in passing—in a published

paper. Meanwhile, in France, Serre was actively

contributing to the false attribution. He made every

effort to associate the name of Andre Weil with the

conjecture, instead of that of Goro Shimura.

"An Exercise for the Interested Reader"

In 1967, Andre Weil wrote a paper in German, in

which he said:19

Ob sich diese Dinge immer, d.h. fur jede iiber Q

definierte Kurve C, so verhalten, scheint im

Moment noch problematisch zu sein und mag dem

interessierten Leser als Ubungsaufgabe empfohlen

werden. x

["Whether these things, that is for every curve C

defined over Q, so behave, at this moment is still

seen as problematic and will be recommended as an

exercise for the interested reader."] This paragraph

refers to elliptic curves over the rational numbers

(which mathematicians denote by Q), and "sich so

verhalten" here refers to being modular, that is, it

states Shimura's conjecture. But, here again, Weil

did not attribute the theory to its originator. He did

so only 12 years later, and even then as "Shimura

asked me..." as we have just seen. In this

p a p e r i n G e r m a n , a b o v e , W e i l c a l l s t h e c o n j e c t u r e " p r o b l e m a t i c . " A n d t h e n h e d o e s s o m e t h i n g s t r a n g e . H e s i m p l y a s s i g n s t h e c o n j e c t u r e a s a n exercise for the interested

reader ( " u n d m a g d e m i n t e r e s s i e r t e n L e s e r a l s U b u n g s a u f g a b e e m p f o h l e n w e r d e n " ) . T h i s e x e r c i s e f o r t h e " i n t e r e s t e d r e a d e r " w o u l d t a k e o n e o f t h e w o r l d ' s f i n e s t m a t h e m a t i c i a n s s e v e n y e a r s o f w o r k i n s o l i t u d e t o a t t e m p t t o p r o v e . W h e n a m a t h e m a t i c i a n a s s i g n s a h o m e w o r k p r o b l e m (Ubungsaufgabe),

u s u a l l y h e o r s h e k n o w s t h e p r o o f t h r o u g h a n d t h r o u g h , a n d b e l i e v e s — k n o w s f o r c e r -t a i n — t h a t t h e t h e o r e m i s t r u e , n o t " p r o b l e m a t i c " a s W e i l d e s c r i b e s i t .

T h e r e i s a n o l d s t o r y a b o u t a m a t h p r o f e s s o r w h o t e l l s h i s c l a s s " t h i s i s o b v i o u s , " w h e n r e f e r r i n g t o s o m e c o n c e p t . T h e c l a s s l o o k s c o n f u s e d s i n c e i t i s n o t a t a l l o b v i o u s , a n d f i n a l l y a b o l d s t u d e n t r a i s e s a h a n d a n d a s k s , " W h y ? " T h e p r o f e s s o r t h e n s t a r t s d r a w i n g a n d w r i t i n g o n t h e e d g e o f t h e b o a r d w i t h o n e h a n d , c o v e r i n g t h e w r i t i n g w i t h h i s o t h e r h a n d , a n d e r a s i n g e v e r y t h i n g a s h e i s d o n e . A f t e r a b o u t t e n m i n u t e s o f t h i s f u r t i v e s c r i b b l i n g , t h e p r o f e s s o r e r a s e s t h e b o a r d c o m p l e t e l y a n d a n n o u n c e s t o t h e b e f u d d l e d c l a s s : " Y e s , i t ' s

o b v i o u s . "

Tbe Lie

I n t h e 1970s, T a n i y a m a ' s p r o b l e m s f r o m t h e T o k y o -N i k k o m e e t i n g r e c e i v e d w i d e r d i s t r i b u t i o n . I n t h e m e a n t i m e , s i n c e W e i l h a d w r i t t e n a b o u t t h e c o n j e c t u r e h e d o u b t e d , m o d u l a r e l l i p t i c c u r v e s b e c a m e k n o w n a s " W e i l C u r v e s . " W h e n T a n i y a m a ' s p r o b l e m s b e c a m e b e t t e r k n o w n i n t h e W e s t , t h e c o n j e c t u r e a b o u t s u c h c u r v e s c a m e t o b e c a l l e d t h e " T a n i y a m a - W e i l c o n j e c t u r e . " S h i m u r a ' s n a m e w a s l e f t o u t c o m p l e t e l y . B u t

s i n c e T a n i y a m a ' s n a m e c a m e i n , W e i l s t a r t e d t o i n v e i g h a g a i n s t c o n j e c t u r e s a l t o g e t h e r . I n 1979, a m e r e f i v e y e a r s b e f o r e i t w a s p r o v e d b y G e r d F a l t i n g s , W e i l e v e n s p o k e a g a i n s t " t h e s o - c a l l e d ' M o r d e l l c o n j e c t u r e ' o n D i o p h a n t i n e e q u a t i o n s . " H e c o n t i n u e d , " I t w o u l d b e n i c e i f t h i s w e r e s o , a n d I w o u l d r a t h e r b e t f o r i t t h e n a g a i n s t i t . B u t i t i s n o m o r e t h a n w i s h f u l t h i n k i n g b e c a u s e t h e r e i s n o t a s h r e d o f e v i d e n c e f o r i t , a n d a l s o n o n e a g a i n s t i t . " B u t W e i l w a s w r o n g e v e n t h e n . A n u m b e r o f R u s s i a n m a t h e m a t i c i a n s , a m o n g t h e m S h a f a r e v i c h a n d P a r s h i n , w e r e a l r e a d y o b t a i n i n g r e s u l t s t h a t w o u l d s u p p l y e v i d e n c e f o r t h e M o r d e l l c o n j e c t u r e a s e a r l y a s t h e e a r l y 1970s. I n 1984, o f c o u r s e , G e r d F a l t i n g s w o u l d p r o v e t h e c o n j e c t u r e o u t r i g h t , m a k i n g F e r m a t ' s L a s t T h e o r e m " a l m o s t a l w a y s t r u e . "

W h i l e A n d r e W e i l w a s t u r n i n g a g a i n s t a l l c o n j e c t u r e s a t a t i m e w h e n h i s n a m e w a s n o l o n g e r b e i n g e x c l u s i v e l y a s s o c i a t e d w i t h t h e c o n j e c t u r e n o w c a l l e d T a n i y a m a - W e i l b y m a n y m a t h e m a t i c i a n s , S e r r e i n P a r i s w a s w o r k i n g t o k e e p S h i m u r a ' s n a m e d i s s o c i a t e d f r o m t h e c o n j e c t u r e . I n 1986,

a t a p a r t y a t t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y , a n d w i t h i n e a r s h o t o f a

n u m b e r o f p e o p l e , Jean- P i e r r e S e r r e t o l d S e r g e L a n g t h a t A n d r e W e i l h a d t o l d h i m o f a c o n v e r s a t i o n h e h a d h a d with

G o r o S h i m u r a . A c c o r d i n g t o S e r r e , this i s what W e i l t o l d h i m t o o k p l a c e :

Weil: Why did Taniyama think that all elliptic

curves are modular?

Shimura: You told him so yourself, and you have

forgotten.

A t this m o m e n t , L a n g , w h o u n k n o w i n g l y h a d b e e n u s i n g t h e t e r m s "Weil c u r v e " a n d " T a n i y a m a -Weil c o n j e c t u r e , " b e c a m e

s u s p i c i o u s . H e t o o k i t u p o n h i m s e l f t o f i n d t h e t r u t h . L a n g i m m e d i a t e l y w r o t e t o b o t h W e i l a n d S h i m u r a , t h e n t o S e r r e . S h i m u r a c a t e g o r i c a l l y d e n i e d t h a t s u c h a c o n v e r s a t i o n e v e r t o o k p l a c e , a n d g a v e a m p l e e v i d e n c e f o r t h i s c l a i m . W e i l d i d n o t r e p l y r i g h t a w a y . A n d S e r r e , i n h i s r e s p o n s e , c r i t i c i z e d L a n g ' s a t t e m p t t o f i n d t h e t r u t h . I n h i s B o u r b a k i S e m i n a r i n J u n e 1995,

S e r r e s t i l l r e f e r r e d t o t h e c o n j e c t u r e a s t h a t o f " T a n i y a m a - W e i l , " l e a v i n g o u t t h e n a m e o f i t s o r i g i n a t o r , w h o t r u s t e d h i m w i t h h i s c o n j e c t u r e 3 0 y e a r s e a r l i e r . W e i l r e s p o n d e d a f t e r a s e c -o n d a t t e m p t t o c o n t a c t h i m b y L a n g . H i s l e t t e r f o l l o w s . 2 0

3 December 1986.

Dear Lang,

I do not recall when and where your letter of

August 9 first reached me. When it did, I had (and

still have) far more serious matters to think about.

I cannot but resent strongly any suggestion that

I ever sought to diminish the credit due to

Taniyama and to Shimura. I am glad to see that

you admire them. So do 1.

Reports of conversations held long ago are open

to misunderstandings. You choose to regard them

as "history",- they are not. At best they are

anecdotes. Concerning the controversy which you

have found fit to raise, Shimura's letters seem to

me to put an end to it, once and for all.

As to attaching names to concepts, theorems, or

(?) conjectures, I have often said: (a) that, when a

proper name gets attached to (say) a concept, this

should never be taken as a sign that the author in

question had anything to do with the concept;

more often than not, the opposite is true.

Pythagoras had

nothing to do with "his" theorem, nor Fuchs with the Fonctions

fuchsiennes, any more than Auguste Comte with rue Auguste-

Comte,- (b) proper names tend, quite properly, to get replaced

by more appropriate ones,- the Leray-Koszul sequence is now a

spectral sequence (and as Siegel once told Erdos, abelian is now

written with a small a).

Why shouldn't I have made "stupid" remarks sometimes, as

you are pleased to say? But indeed, 1 was "out of it" in 1979

when expressing some skepticism about Mordell's conjecture,

since at that time 1 was totally ignorant of the work of the

Russians (Parshin, etc.) in that direction. My excuse, if it is one,

is that I had had long conversations with Shafarevich in 1972,

and he never mentioned any of that work.

Sincerely,

A. Weil

AW:ig

RS. Should you wish to run this letter through your Xerox

machine, do feel free to do so. I wonder what the Xerox Co.

would do without you and the like of you.

Deep in the Black Forest, Fall i 984

W h i l e t h e c o n t r o v e r s y a b o u t w h o o r i g i n a t e d t h e S h i m u r a -T a n i y a m a c o n j e c t u r e w a s r a g i n g i n B e r k e l e y , N e w H a v e n , P r i n c e t o n , a n d a c r o s s t h e A t l a n t i c i n P a r i s , s o m e t h i n g t o t a l l y u n e x p e c t e d w a s h a p p e n i n g d e e p i n t h e B l a c k F o r e s t o f s o u t h w e s t G e r m a n y .

G e r h a r d F r e y r e c e i v e d h i s D i p l o m a f r o m t h e U n i v e r s i t y o f T u b i n g e n , a n d h i s P h . D . f r o m t h e U n i v e r s i t y o f H e i d e l b e r g ,

w h e r e h e s t u d i e d n u m b e r t h e o r y a n d w a s i n f l u e n c e d b y t h e w o r k s o f H a s s e a n d W e i l . F r e y w a s f a s c i n a t e d b y t h e i n t e r p l a y b e t w e e n t h e t h e o r y o f n u m b e r s a n d a l g e b r a i c g e o m e t r y , a n a r e a o f m a t h e m a t i c s w h i c h w a s d e v e l o p e d i n t h e l a s t f i f t y y e a r s . H e w a s a l s o i n t e r e s t e d i n a r i t h m e t i c g e o m e t r y . I t w a s t h e c o n n e c t i o n s b e t w e e n n u m b e r t h e o r y a n d t h e n e w e r f i e l d s o f a l g e b r a i c a n d a r i t h m e t i c g e o m e t r y t h a t w o u l d l e a d h i m t o m a k e a n u n e x p e c t e d m a t h e m a t i c a l s t a t e m e n t . I n t h e 1970s, F r e y d i d a l o t o f w o r k o n e l l i p t i c c u r v e s a n d D i o p h a n t i n e e q u a t i o n s , a n d i n 1978 h e r e a d t h e p a p e r " M o d u l a r c u r v e s a n d t h e E i s e n s t e i n i d e a l , " b y B a r r y M a z u r o f H a r v a r d U n i v e r s i t y . F r e y w a s s t r o n g l y i n f l u e n c e d b y t h e p a p e r , a s w e r e m a n y n u m b e r t h e o r i s t s , a m o n g t h e m B e r k e l e y ' s K e n n e t h R i b e t a n d P r i n c e t o n ' s A n d r e w W i l e s . F r e y b e c a m e c o n v i n c e d b y M a z u r ' s p a p e r t h a t h e s h o u l d t h i n k v e r y s e r i o u s l y a b o u t a p p l i c a t i o n s o f m o d u l a r c u r v e s a n d G a l o i s r e p r e s e n t a t i o n s t o t h e t h e o r y o f e l l i p t i c c u r v e s . H e f o u n d t h a t t h i s l e d h i m

a l m o s t u n a v o i d a b l y t o D i o p h a n t i n e q u e s t i o n s c l o s e l y r e l a t e d t o e q u a t i o n s o f F e r m a t ' s t y p e . T h i s w a s a p o w e r f u l i n s i g h t , w h i c h F r e y t r i e d t o m a k e m o r e p r e c i s e .

I n 1981, G e r h a r d F r e y s p e n t a f e w w e e k s a t H a r v a r d U n i -v e r s i t y a n d h a d a n u m b e r o f d i s c u s s i o n s w i t h B a r r y M a z u r . T h e s e d i s c u s s i o n s w e r e c l e a r i n g t h i n g s i n h i s m i n d . T h e h e a v y f o g s u r r o u n d i n g t h e d i f f i c u l t c o n n e c t i o n s h e e n v i s i o n e d b e t w e e n F e r m a t - l i k e e q u a t i o n s a n d t h e r e l a t i o n b e t w e e n m o d u l a r f o r m s a n d e l l i p t i c c u r v e s w a s s l o w l y l i f t i n g . F r e y w e n t o n t o B e r k e l e y , w h e r e h e s p o k e w i t h K e n R i b e t , a b r i g h t n u m b e r t h e o r i s t w h o w a s a g r a d u a t e o f H a r v a r d a n d h a d w o r k e d w i t h M a z u r o n r e l a t e d i s s u e s . F r e y r e t u r n e d t o h i s n a t i v e G e r m a n y .

no

T h r e e y e a r s l a t e r , h e w a s i n v i t e d t o g i v e a l e c t u r e a t t h e O b e r - w o l f a c h c e n t e r d e e p i n t h e B l a c k F o r e s t .

O b e r w o l f a c h w a s d e s i g n e d a s a c o n f e r e n c e a n d w o r k s h o p c e n t e r i n m a t h e m a t i c s , s e t i n b e a u t i f u l a n d p e a c e f u l s u r r o u n d i n g s f a r f r o m c i t i e s a n d c r o w d s . E v e r y y e a r , t h e c e n t e r s p o n s o r s a b o u t f i f t y i n t e r n a t i o n a l m e e t i n g s o n d i f f e r e n t t o p i c s o f m a t h e -m a t i c s . L e c t u r e s , a n d e v e n j u s t a t t e n d a n c e a t t h e m e e t i n g s , a r e e x c l u s i v e l y b y i n v i t a t i o n . E v e r y e f f o r t i s m a d e t o a l l o w f o r t h e e a s y e x c h a n g e o f i d e a s a m o n g e x p e r t s f r o m d i f f e r e n t c o u n -t r i e s . I n 1984, G e r h a r d F r e y g a v e a t a l k a t a n u m b e r t h e o r y c o n f e r e n c e t h e r e . H e m a d e w h a t l o o k e d l i k e a c r a z y a s s e r t i o n . T h e m i m e o g r a p h e d s h e e t s f i l l e d w i t h m a t h e m a t i c a l f o r m u l a s h e p a s s e d a r o u n d a t t h e c o n f e r e n c e s e e m e d t o i m p l y t h a t i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w e r e i n d e e d t r u e , F e r m a t ' s L a s t T h e o r e m w o u l d b e p r o v e d . T h i s m a d e n o s e n s e a t a l l . W h e n K e n R i b e t f i r s t h e a r d o f F r e y ' s s t a t e m e n t , h e t h o u g h t i t w a s a j o k e . W h a t c o u l d m o d u l a r i t y o f e l l i p t i c c u r v e s p o s s i b l y h a v e t o d o w i t h F e r m a t ' s L a s t T h e o r e m ? h e a s k e d h i m s e l f . H e g a v e t h i s s t r a n g e a s s e r t i o n n o f u r t h e r t h o u g h t a n d w e n t a b o u t h i s u s u a l w o r k . B u t a n u m b e r o f p e o p l e i n P a r i s a n d

e l s e w h e r e w e r e i n t e r e s t e d i n F r e y ' s u n p r o v e n , a n d s o m e w h a t i n c o m p l e t e s t a t e m e n t . J e a n - P i e r r e S e r r e w r o t e a p r i v a t e l e t t e r t o a m a t h e m a t i c i a n b y t h e n a m e o f J . - F . M e s t r e . T h i s l e t t e r b e c a m e p u b l i c , a n d S e r r e s u b s e q u e n t l y p u b l i s h e d a p a p e r r e p e a t i n g h i s o w n c o n j e c t u r e s f r o m t h e l e t t e r t o M e s t r e . 2 1

Ribet's Theorem

K e n R i b e t , w h o f i r s t t h o u g h t t h i s s t a t e m e n t w a s a j o k e , s t a r t e d t h i n k i n g a b o u t S e r r e ' s c o n j e c t u r e s , a n d i n f a c t r e c o g n i z e d i n

t h e m s o m e t h i n g h e h a d a l r e a d y f o r m u l a t e d f o r h i m s e l f w h e n h e f o u n d t i m e t o t h i n k a b o u t F r e y ' s " j o k e . " T h e s e w e r e c e r t a i n c l a r i f i c a t i o n s o f G e r h a r d F r e y ' s s t a t e m e n t s , w h i c h , i f p r o v e n , w o u l d e s t a b l i s h t h e f o l l o w i n g i m p l i c a t i o n :

Shimura-Taniyama conjecture > Fermat's Last Theorem

T h e w a y t h e F r e y i d e a w o r k e d w a s i n g e n i o u s . F r e y r e a s o n e d a s f o l l o w s : S u p p o s e t h a t F e r m a t ' s L a s t T h e o r e m i s not t r u e . T h e n , f o r s o m e p o w e r n t h a t i s g r e a t e r t h a n 1 t h e r e is a s o l u t i o n t o F e r m a t ' s e q u a t i o n : x n + yn = z",

w h e r e x , y, a n d z a r e i n t e g e r s . T h i s p a r t i c u l a r s o l u t i o n , a , b , a n d c, w o u l d t h e n g i v e r i s e t o a s p e c i f i c e l l i p t i c c u r v e . N o w F r e y w r o t e ' d o w n t h e g e n e r a l e q u a t i o n o f t h i s c u r v e t h a t w o u l d r e s u l t f r o m t h e s o l u t i o n o f F e r m a t ' s e q u a t i o n . H i s c o n j e c t u r e p r e s e n t e d a t O b e r w o l f a c h s t a t e d t h a t t h i s v e r y c u r v e , n o w c a l l e d t h e F r e y c u r v e , w a s a v e r y s t r a n g e a n i m a l . I t w a s s o s t r a n g e , i n f a c t , t h a t i t c o u l d n ' t p o s s i b l y e x i s t . A n d , m o s t i m p o r t a n t , t h e e l l i p t i c c u r v e t h a t w o u l d a r i s e i f F e r m a t ' s L a s t T h e o r e m w e r e f a l s e w a s d e f i n i t e l y not m o d u l a r . S o , i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s i n d e e d t r u e ,

t h e n a l l e l l i p t i c c u r v e s m u s t b e m o d u l a r . T h e r e f o r e , a n e l l i p t i c c u r v e t h a t w a s n o t m o d u l a r c o u l d n ' t p o s s i b l y e x i s t . A n d i t w o u l d f o l l o w t h a t F r e y ' s c u r v e , a n e l l i p t i c c u r v e t h a t w a s n o t m o d u l a r ( i n a d d i t i o n t o a l l i t s o t h e r s t r a n g e c h a r a c t e r i s t i c s ) c o u l d n o t e x i s t . T h e r e f o r e , t h e s o l u t i o n s t o F e r m a t ' s e q u a t i o n c o u l d n o t e x i s t e i t h e r . W i t h o u t t h e e x i s t e n c e o f s o l u t i o n s t o t h e F e r m a t e q u a t i o n , F e r m a t ' s L a s t T h e o r e m ( w h i c h s t a t e s t h a t t h e r e a r e no s o l u t i o n s t o t h e e q u a t i o n f o r a n y n>2),

w o u l d b e p r o v e d . T h i s w a s a c o m p l i c a t e d s e q u e n c e o f i m p l i c a t i o n s , b u t i t f o l l o w e d b e a u t i f u l l y t h e l o g i c o f m a t h e m a t i c a l

Ken Ribet presenting his important

theorems.

Andrew Wiles being

interviewed.

Barry Mazur of Harvard

University— the "granddaddy" of

them all, a mathematician whose

geometrical insights inspired

everyone who contributed to the

final proof of Fermat's Last

Theorem.

Gerhard Frey, who had the

"crazy idea" that an elliptic

curve resulting from a

solution of Fermat's

equation simply could not

exist.

p r o o f . T h e l o g i c w a s : A i m p l i e s B ; t h e r e f o r e , i f B i s not t r u e , t h e n A c a n n o t b e t r u e e i t h e r . H o w e v e r , t h e F r e y s t a t e m e n t w a s , i t s e l f , a conjecture. I t w a s a c o n j e c t u r e w h i c h s a i d t h a t i f a n o t h e r c o n j e c t u r e ( S h i m u r a -T a n i y a m a ) w a s t r u e , t h e n F e r m a t ' s L a s t T h e o r e m w o u l d b e e s t a b l i s h e d . T h e p a i r o f s u b s e q u e n t c o n j e c t u r e s i n S e r r e ' s l e t t e r t o M e s t r e f u r t h e r a l l o w e d K e n R i b e t t o t h i n k a b o u t t h e F r e y c o n j e c t u r e i n c l e a r t e r m s .

K e n R i b e t h a d n e v e r b e f o r e b e e n i n t e r e s t e d i n F e r m a t ' s L a s t T h e o r e m . H e h a d s t a r t e d o u t a s a c h e m i s t r y m a j o r a t B r o w n U n i v e r s i t y . U n d e r t h e i n f l u e n c e a n d t u t e l a g e o f K e n n e t h F . I r e -l a n d , R i b e t w a s s t e e r e d t o m a t h e m a t i c s a n d g o t i n t e r e s t e d i n z e t a f u n c t i o n s , e x p o n e n t i a l s u m s , a n d n u m b e r t h e o r y . H e h a d d i s m i s s e d F e r m a t ' s L a s t T h e o r e m a s " o n e o f t h o s e p r o b l e m s a b o u t w h i c h n o t h i n g f u r t h e r o f r e a l i m p o r t a n c e c o u l d b e s a i d . " T h i s w a s a v i e w h e l d b y m a n y m a t h e m a t i c i a n s , b e c a u s e p r o b l e m s i n n u m b e r t h e o r y t e n d t o b e i s o l a t e d ,

w i t h n o u n i f y i n g s c h e m e o r u n d e r l y i n g g e n e r a l p r i n c i p l e b e h i n d t h e m . W h a t i s i n t e r e s t i n g a b o u t F e r m a t ' s L a s t T h e o r e m , h o w e v e r , i s t h a t i t s p a n s m a t h e m a t i c a l h i s t o r y f r o m t h e d a w n o f c i v i l i z a t i o n t o o u r o w n t i m e . A n d t h e t h e o r e m ' s u l t i m a t e s o l u t i o n a l s o s p a n s t h e b r e a d t h o f m a t h e m a t i c s , i n v o l v i n g f i e l d s o t h e r t h a n n u m b e r t h e o r y : a l g e b r a , a n a l y s i s , g e o m e t r y , a n d t o p o l o g y — v i r t u a l l y a l l o f m a t h e m a t i c s .

R i b e t w e n t o n t o p u r s u e a P h . D . i n m a t h e m a t i c s a t H a r v a r d U n i v e r s i t y . T h e r e , f i r s t i n d i r e c t l y a n d t o w a r d h i s g r a d u a t i o n m o r e d i r e c t l y , h e f e l l u n d e r t h e i n f l u e n c e o f t h e g r e a t n u m b e r t h e o r i s t a n d g e o m e t e r B a r r y M a z u r , w h o s e v i s i o n i n s p i r e d e v e r y m a t h e m a t i c i a n i n v o l v e d e v e n i n t h e s m a l l e s t w a y i n e f f o r t s t o p r o v e F e r m a t ' s L a s t T h e o r e m . M a z u r ' s p a p e r o n t h e

I 14

Eisenstein ideal acted as an abstraction of the theory

of ideals developed in the last century by Ernst

Kummer into the modern fields of mathematics,

algebraic geometry and new approaches to the

theory of numbers through geometry.22

Ken Ribet eventually became a professor of

mathematics at the University of California at

Berkeley and did research in number theory. In 1985,

he heard about Frey's "crazy" notion that if a solution

of Fermat's equation existed, that is, if Fermat's Last

Theorem were false, then this solution would give

rise to a very weird curve. This Frey Curve would be

associated with an elliptic curve that could not be

modular. The pair of associated conjectures in

Serre's letter to Mestre made him interested in trying

to prove Frey's conjecture. While he wasn't really

interested in Fermat's Last Theorem, Ribet

recognized that this had become a hot problem, and

it happened to be in an area he knew well. During the

week of August 18-24, 1985, Ribet was at a meeting

on arithmetic algebraic geometry in Areata,

California. He began thinking about Frey's statement,

and the problem remained on his mind for the next

year. When he was freed of his teaching obligations

at Berkeley early in the summer of 1986, Ribet flew

to Germany where he was to do research at the

world-famous center for mathematics, the Max

Planck Institute. Just as he arrived at the Institute,

Ribet made his great breakthrough. He was now

almost able to prove Frey's conjecture.

But he was not quite there. When he returned to

Berkeley, Ribet ran into Barry Mazur, who was

visiting from Harvard. "Barry, let's go for a cup of

coffee," Ribet suggested. The two retreated to a

popular cafe by the University of California campus.

While sipping a cappuccino, Ribet confided to

M a z u r : " I ' m t r y i n g t o g e n e r a l i z e w h a t I ' v e d o n e , s o t h a t I ' l l b e a b l e t o p r o v e t h e F r e y c o n j e c t u r e . I j u s t d o n ' t s e e m t o g e t t h i s o n e t h i n g t o g e n e r a l i z e i t . . . " M a z u r l o o k e d a t w h a t h e w a s s h o w i n g h i m . " B u t y o u ' v e d o n e i t a l r e a d y K e n , " h e s a i d , " a l l y o u n e e d t o d o i s a d d o n s o m e e x t r a g a m m a z e r o o f N s t r u c t u r e , a n d r u n t h r o u g h y o u r a r g u m e n t , a n d y o u ' r e t h e r e ! " R i b e t l o o k e d a t M a z u r , h e l o o k e d b a c k a t h i s c a p p u c c i n o , t h e n b a c k a t M a z u r w i t h d i s b e l i e f . " M y G o d , y o u ' r e a b s o l u t e l y r i g h t ! " h e s a i d . L a t e r h e w e n t b a c k t o h i s o f f i c e t o f i n i s h o f f t h e p r o o f . " K e n ' s i d e a w a s b r i l l i a n t , " M a z u r e x c l a i m e d w h e n d e s c r i b i n g K e n R i b e t ' s i n g e n i o u s p r o o f a f t e r i t w a s p u b l i s h e d a n d b e c a m e k n o w n i n t h e w o r l d o f m a t h e m a t i c i a n s .

R i b e t f o r m u l a t e d a n d p r o v e d a t h e o r e m w h i c h e s t a b l i s h e d a s f a c t t h a t i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s t r u e , t h e n F e r m a t ' s L a s t T h e o r e m w o u l d f a l l o u t o f i t a s a d i r e c t c o n s e q u e n c e . T h e m a n w h o o n l y a y e a r e a r l i e r t h o u g h t t h a t F r e y ' s s u g g e s t i o n w a s a j o k e n o w p r o v e d t h a t t h e " j o k e " w a s a c t u a l l y a m a t h e m a t i c a l r e a l i t y . T h e d o o r t o t h e a t t a c k o n F e r m a t ' s p r o b l e m u s i n g t h e m o d e r n m e t h o d s o f a r i t h m e t i c a l g e b r a i c g e o m e t r y w a s n o w w i d e o p e n . A l l t h e w o r l d n e e d e d

n o w w a s s o m e o n e w h o w o u l d p r o v e t h e s e e m i n g l y -i m p o s s i b l e S h i m u r a - T a n i y a m a c o n j e c t u r e . T h e n F e r m a t ' s L a s t T h e o r e m w o u l d a u t o m a t i c a l l y b e t r u e .

A Child's Dream

T h e p e r s o n w h o w a n t e d d o j u s t t h a t w a s A n d r e w W i l e s . W h e n h e w a s t e n y e a r s o l d , A n d r e w W i l e s w e n t t o t h e p u b l i c l i b r a r y i n h i s t o w n i n E n g l a n d a n d l o o k e d a t a b o o k o n m a t h e m a t i c s . I n t h a t b o o k h e r e a d a b o u t F e r m a t ' s L a s t T h e o r e m . T h e t h e o r e m , a s d e s c r i b e d i n t h e b o o k , s e e m e d s o s i m p l e , t h a t a n y

c h i l d c o u l d u n d e r s t a n d i t . I n W i l e s ' o w n w o r d s : " I t s a i d t h a t y o u w i l l n e v e r f i n d n u m b e r s , x, y , a n d z , s o t h a t x3 + y 3 = z 3 . N o m a t t e r h o w h a r d y o u t r i e d , y o u w i l l n e v e r , e v e r f i n d s u c h n u m b e r s . A n d i t s a i d t h a t t h e s a m e w a s t r u e f o r x4 + y 4 = z 4 , a n d f o r x5

+ y5 = z 5 , a n d s o o n . . . I t s e e m e d s o s i m p l e . A n d i t s a i d t h a t n o b o d y h a s e v e r f o u n d a p r o o f o f t h i s f o r o v e r t h r e e h u n d r e d y e a r s . I w a n t e d t o p r o v e i t . . . "

I n t h e 1970s, A n d r e w W i l e s w e n t t o t h e u n i v e r s i t y . W h e n h e f i n i s h e d h i s d e g r e e h e w a s a d m i t t e d a s a r e s e a r c h s t u d e n t i n m a t h e m a t i c s t o C a m b r i d g e . H i s a d v i s e r w a s P r o f e s s o r J o h n C o a t e s . W i l e s h a d t o d r o p h i s c h i l d h o o d d r e a m o f p r o v i n g F e r m a t ' s L a s t T h e o r e m . R e s e a r c h o n t h e p r o b l e m w o u l d h a v e t u r n e d i n t o s u c h a w a s t e o f t i m e t h a t n o g r a d u a t e s t u d e n t c o u l d a f f o r d i t . B e s i d e s , w h a t d o c t o r a l a d v i s e r w o u l d h a v e a c c e p t e d a s t u d e n t w o r k i n g o n s u c h a n a n c i e n t p u z z l e , o n e t h a t h a d k e p t t h e w o r l d ' s b r i g h t e s t m i n d s f r o m a s o l u t i o n f o r t h r e e c e n t u r i e s ? I n t h e 1970s,

F e r m a t w a s n o t i n f a s h i o n . W h a t w a s " i n " a t t h e t i m e , t h e r e a l h o t t o p i c f o r r e s e a r c h i n n u m b e r t h e o r y , w a s e l l i p t i c c u r v e s . S o A n d r e w W i l e s s p e n t h i s t i m e d o i n g r e s e a r c h o n e l l i p t i c c u r v e s a n d i n a n a r e a c a l l e d I w a s a w a t h e o r y . H e c o m p l e t e d h i s d o c t o r a l d i s s e r t a t i o n , a n d w h e n h e w a s a w a r d e d h i s P h . D . , h e

g o t a p o s i t i o n i n m a t h e m a t i c s a t P r i n c e t o n U n i v e r s i t y a n d m o v e d t o t h e U n i t e d S t a t e s . T h e r e , h e c o n t i n u e d d o i n g r e s e a r c h o n e l l i p t i c c u r v e s a n d I w a s a w a t h e o r y .

An Old Flame Rekindled

I t w a s a w a r m s u m m e r e v e n i n g , a n d A n d r e w W i l e s w a s s i p p i n g i c e d t e a a t a f r i e n d ' s h o u s e . S u d d e n l y , i n t h e m i d d l e o f t h e c o n -v e r s a t i o n , h i s f r i e n d s a i d : " B y t h e w a y , d i d y o u h e a r t h a t K e n

R i b e t j u s t p r o v e d t h e E p s i l o n C o n j e c t u r e ? " T h e E p s i l o n C o n j e c t u r e i s w h a t n u m b e r t h e o r i s t s w e r e i n f o r m a l l y c a l l i n g F r e y ' s c o n j e c t u r e , a s m o d i f i e d b y S e r r e , a b o u t t h e c o n n e c t i o n b e t w e e n F e r m a t ' s L a s t T h e o r e m a n d t h e S h i m u r a - T a n i y a m a C o n j e c t u r e . W i l e s w a s e l e c t r i f i e d . A t t h a t m o m e n t , h e k n e w t h a t h i s l i f e w a s c h a n g i n g . T h e c h i l d h o o d d r e a m h e h a d o f p r o v i n g F e r m a t ' s L a s t T h e o r e m — a d r e a m h e h a d h a d t o g i v e u p t o u n d e r t a k e m o r e f e a s i b l e r e s e a r c h — c a m e a l i v e a g a i n w i t h i n c r e d i b l e f o r c e . H e w e n t h o m e a n d s t a r t e d t h i n k i n g a b o u t h o w h e w o u l d p r o v e t h e S h i m u r a - T a n i y a m a c o n j e c t u r e .

" F o r t h e f i r s t f e w y e a r s , " h e l a t e r c o n f i d e d , "I k n e w I

h a d n o c o m p e t i t i o n , s i n c e I

k n e w t h a t n o b o d y — m e i n c l u d e d — h a d a n y i d e a w h e r e t o s t a r t . " H e d e c i d e d t o w o r k i n c o m p l e t e s e c r e c y , a n d i n i s o l a t i o n . " T o o m a n y s p e c t a t o r s w o u l d s p o i l t h e c o n c e n t r a t i o n , a n d I

d i s c o v e r e d e a r l y o n t h a t j u s t a m e n t i o n o f F e r m a t i m m e d i a t e l y g e n e r a t e s t o o m u c h i n t e r e s t . " O f c o u r s e , g i f t e d , a b l e m a t h e m a t i c i a n s a b o u n d , e s p e c i a l l y a t a p l a c e l i k e P r i n c e t o n , a n d t h e d a n g e r o f s o m e o n e c o m p l e t i n g y o u r w o r k f o r y o u— a n d e v e n d o i n g i t b e t t e r — i s v e r y r e a l .

W h a t e v e r t h e r e a s o n , W i l e s l o c k e d h i m s e l f u p i n h i s a t t i c o f f i c e a n d w e n t t o w o r k . H e a b a n d o n e d a l l o t h e r

r e s e a r c h p r o j e c t s t o d e v o t e h i s t i m e c o m p l e t e l y t o F e r m a t . W i l e s w o u l d u s e a l l t h e p o w e r o f t h e m o d e r n m a c h i n e r y o f a l g e b r a , g e o m e t r y , a n a l y s i s , a n d t h e o t h e r a r e a s o f m a t h e m a t i c s . H e w o u l d a l s o m a k e u s e o f t h e i m p o r t a n t m a t h e m a t i c a l r e s u l t s o f h i s c o n t e m p o -r a r i e s , a n d o f h i s h i s t o r i c a l p r e d e c e s s o r s . H e w o u l d m a k e u s e o f R i b e t ' s c l e v e r m e t h o d s o f p r o o f , a n d h i s r e s u l t s , - h e w o u l d u s e t h e t h e o r i e s o f B a r r y M a z u r , a n d t h e i d e a s o f S h i m u r a , F r e y , S e r r e , A n d r e W e i l , a n d t h o s e o f m a n y , m a n y o t h e r m a t h e m a t i c i a n s .

W i l e s ' g r e a t n e s s , G e r h a r d F r e y w o u l d l a t e r s a y , w a s t h a t h e b e l i e v e d i n w h a t h e w a s d o i n g a t a t i m e w h e n v i r t u a l l y e v e r y m a t h e m a t i c i a n i n t h e w o r l d b e l i e v e d t h a t t h e S h i m u r a -T a n i y a m a c o n j e c t u r e c o u l d n o t b e p r o v e n i n t h e t w e n t i e t h c e n t u r y .

To p r o v e t h e S h i m u r a -T a n i y a m a C o n j e c t u r e , A n d r e w W i l e s k n e w h e h a d t o p r o v e t h a t e v e r y e l l i p t i c c u r v e i s m o d u l a r . H e h a d t o s h o w t h a t e v e r y e l l i p t i c c u r v e , w h o s e s o l u t i o n s l i e o n a d o u g h n u t , w a s r e a l l y a m o d u l a r f o r m i n d i s g u i s e . T h e d o u g h n u t w a s , i n a s e n s e , a l s o t h i s s p a c e o f i n t r i c a t e l y s y m m e t r i c f u n c t i o n s o n t h e c o m p l e x p l a n e c a l l e d m o d u l a r f o r m s . N o b o d y h a d a n y i d e a h o w t o s h o w s u c h a w e i r d c o n n e c t i o n b e t w e e n t h e s e t w o s e e m i n g l y v e r y d i f f e r e n t e n t i t i e s .

W i l e s r e a l i z e d t h a t t h e b e s t i d e a w a s t o t r y t o count

t h e n u m b e r o f e l l i p t i c c u r v e s , a n d t o c o u n t t h e n u m b e r o f modular e l l i p t i c c u r v e s , a n d t h e n t o s h o w t h a t t h e i r " n u m b e r " w a s t h e s a m e . T h i s c o n s t r u c t i o n w o u l d p r o v e t h a t t h e e l l i p t i c c u r v e s a n d t h e m o d u l a r e l l i p t i c c u r v e s w e r e o n e a n d t h e s a m e , a n d h e n c e e v e r y e l l i p t i c c u r v e i s i n d e e d m o d u l a r , a s t h e S h i m u r a -T a n i y a m a c o n j e c t u r e c l a i m s .

W i l e s r e a l i z e d t w o t h i n g s . O n e w a s t h a t h e d i d n ' t h a v e t o p r o v e t h e e n t i r e S h i m u r a -T a n i y a m a c o n j e c t u r e , b u t

o n l y a s p e c i a l c a s e : semistable

e l l i p t i c c u r v e s w i t h r a t i o n a l n u m b e r s a s t h e c o e f f i c i e n t s . P r o v i n g t h a t t h e c o n j e c t u r e w a s t r u e f o r t h i s s m a l l e r c l a s s o f e l l i p t i c c u r v e s w o u l d b e e n o u g h t o e s t a b l i s h F e r m a t ' s L a s t T h e o r e m . T h e o t h e r t h i n g W i l e s k n e w w a s t h a t " c o u n t i n g " w o u l d n o t w o r k h e r e b e c a u s e h e w a s d e a l i n g w i t h infinite s e t s . T h e s e t o f s e m i s t a b l e e l l i p t i c c u r v e s w a s i n f i n i t e . A n y r a t i o n a l n u m b e r alb, w h e r e a a n d b a r e i n t e g e r s , w o u l d

g i v e y o u a n o t h e r e l l i p t i c c u r v e ( w e s a y a n e l l i p t i c c u r v e over t h e r a t i o n a l s ) . S i n c e t h e r e a r e i n f i n i t e l y m a n y s u c h n u m b e r s — a a n d b c a n b e a n y o f t h e i n f i n i t e l y m a n y n u m b e r s 1,2,3,4..., t o i n f i n i t y , t h e r e a r e i n f i n i t e l y m a n y e l l i p t i c c u r v e s . S o c o u n t i n g a s w e k n o w i t w o u l d n ' t w o r k .

Breaking Down a Big Problem into Smaller Ones W i l e s t h o u g h t h e m i g h t t r y t o w o r k o n s m a l l e r p r o b l e m s , o n e a t a t i m e . M a y b e h e c o u l d l o o k a t s e t s o f e l l i p t i c f u n c t i o n s a n d s e e w h a t h e c o u l d d o a b o u t t h e m . T h i s w a s a g o o d a p p r o a c h s i n c e i t b r o k e d o w n t h e t a s k s o t h a t , s t e p b y s t e p , h e c o u l d u n d e r s t a n d e a c h s e t . F i r s t o f a l l , s o m e e l l i p t i c c u r v e s w e r e a l r e a d y k n o w n t o b e m o d u l a r . T h e s e w e r e v e r y i m p o r t a n t r e s u l t s , d e v e l o p e d b y m a n y o t h e r n u m b e r t h e o r i s t s . B u t s o o n A n d r e w W i l e s r e a l i z e d t h a t l o o k i n g o n l y a t e l l i p t i c c u r v e s a n d t r y i n g t o c o u n t t h e m o f f a g a i n s t m o d u l a r f o r m s m i g h t n o t b e a g o o d a p p r o a c h — h e w a s d e a l i n g w i t h t w o i n f i n i t e s e t s . I n f a c t , h e w a s n o c l o s e r t o a s o l u t i o n t h a n w a s t h e s k e p t i c a l A n d r e W e i l w h e n h e s a i d : " I s e e n o r e a s o n a g a i n s t t h e c o n j e c t u r e s i n c e o n e a n d t h e o t h e r o f t h e t w o s e t s a r e d e n u m e r a b l e [ i n f i -n i t e b u t o f t h e o r d e r o f i n f i n i t y o f t h e i n t e g e r s a n d r a t i o n a l n u m b e r s , n o t t h e h i g h e r o r d e r o f i n f i n i t y o f t h e i r r a t i o n a l n u m b e r s a n d t h e c o n t i n u u m ] , b u t I d o n ' t s e e a n y r e a s o n f o r i t ,

e i t h e r . . . " A f t e r t w o y e a r s o f g e t t i n g n o w h e r e , W i l e s t r i e d a n e w a p p r o a c h . H e t h o u g h t h e m i g h t transform t h e e l l i p t i c c u r v e s i n t o G a l o i s r e p r e s e n t a t i o n s , a n d t h e n c o u n t t h e s e G a l o i s r e p -r e s e n t a t i o n s a g a i n s t t h e m o d u l a r f o r m s .

T h e i d e a w a s a n e x c e l l e n t o n e , a l t h o u g h i t w a s n o t o r i g i n a l . T h e p r i n c i p l e b e h i n d t h i s m o v e i s i n t e r e s t i n g . N u m b e r t h e o r i s t s

a r e c o n c e r n e d w i t h f i n d i n g s o l u t i o n s o f e q u a t i o n s , s u c h a s t h e F e r m a t e q u a t i o n . T h e m a t h e m a t i c a l t h e o r y o f f i e l d s o f n u m b e r s s e t s t h i s p r o b l e m i n t h e c o n t e x t o f f i e l d e x t e n s i o n s . F i e l d s a r e l a r g e , i n f i n i t e c o l l e c t i o n s w h i c h a r e d i f f i c u l t t o a n a l y z e . T h e r e f o r e , w h a t n u m b e r t h e o r i s t s h a v e o f t e n d o n e i s t o u s e t h e t h e o r i e s o f E v a r i s t e G a l o i s , c a l l e d G a l o i s t h e o r y , i n o r d e r t o t r a n s l a t e t h e s e p r o b l e m s f r o m t h e c o m p l i c a t e d f i e l d s t o w h a t a r e k n o w n a s g r o u p s . O f t e n a g r o u p i s g e n e r a t e d b y a f i n i t e ( r a t h e r t h a n a n i n f i n i t e ) s e t o f e l e m e n t s . U s i n g G a l o i s t h e o r y t h u s a l l o w s t h e n u m b e r t h e o r i s t t o m o v e f r o m a n i n f i n i t e c o l l e c t i o n t o o n e t h a t i s r e p r e s e n t e d b y a f i n i t e s e t . T h i s t r a n s l a t i o n o f a p r o b l e m c o n s t i t u t e s a n i m m e n s e s t e p f o r w a r d , s i n c e a f i n i t e s e t o f e l e m e n t s i s s o m u c h e a s i e r t o h a n d l e t h a n a n i n f i n i t e s e t . C o u n t i n g m a k e s s e n s e o n l y f o r a f i n i t e n u m b e r o f e l e m e n t s . T h e a p p r o a c h s e e m e d t o w o r k f o r s o m e s e t s o f e l l i p t i c c u r v e s . T h i s w a s a g o o d b r e a k t h r o u g h . B u t a f t e r a n o t h e r y e a r , W i l e s w a s s t u c k a g a i n .

The Flack Paper

W h a t A n d r e w W i l e s w a s t r y i n g t o d o n o w w a s t o c o u n t s e t s o f G a l o i s r e p r e s e n t a t i o n s , c o r r e s p o n d i n g t o t h e ( s e m i -s t a b l e ) e l l i p t i c c u r v e s a g a i n s t t h e m o d u l a r f o r m s ,

a n d t o s h o w t h a t t h e y a r e t h e s a m e . I n d o i n g s o , h e w a s u s i n g h i s a r e a o f e x p e r -t i s e , i n w h i c h h e h a d d o n e h i s d i s s e r t a t i o n , c a l l e d H o r i z o n t a l I w a s a w a T h e o r y . W i l e s w a s t r y i n g t o u s e t h i s t h e o r y t o a t t a i n t h e C l a s s N u m b e r F o r m u l a , a r e s u l t w h i c h h e n e e d e d f o r t h e " c o u n t i n g . " B u t h e r e h e w a s u p a g a i n s t a b r i c k w a l l . N o t h i n g h e c o u l d d o b r o u g h t h i m n e a r e r t o t h e a n s w e r .

I n t h e s u m m e r o f 1991, W i l e s w a s a t a c o n f e r e n c e i n B o s t o n , w h e r e h e m e t h i s f o r m e r d o c t o r a l a d v i s e r a t C a m b r i d g e , J o h n

C o a t e s . P r o f e s s o r C o a t e s t o l d W i l e s t h a t o n e o f C o a t e s ' s t u d e n t s , M a t t h i a s F l a c h , u s i n g e a r l i e r w o r k b y a R u s s i a n m a t h e m a t i c i a n n a m e d K o l y v a g i n , h a d d e v i s e d a n E u l e r S y s t e m ( n a m e d a f t e r L e o n h a r d E u l e r ) i n a n a t t e m p t t o p r o v e t h e C l a s s N u m b e r F o r -m u l a . T h i s w a s e x a c t l y w h a t W i l e s n e e d e d f o r h i s p r o o f o f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e — i f , i n d e e d , h e c o u l d e x t e n d F l a c h ' s p a r t i a l r e s u l t s t o t h e f u l l C l a s s N u m b e r F o r m u l a . W i l e s w a s e l a t e d o n h e a r i n g f r o m C o a t e s a b o u t t h i s w o r k o f F l a c h . " T h i s w a s t a i l o r -m a d e " f o r h i s o w n p r o b l e m , W i l e s s a i d , a s i f M a t t h i a s F l a c h h a d d o n e a l l t h i s w o r k j u s t f o r h i m . A n d W i l e s i m m e d i a t e l y a b a n d o n e d a l l h i s H o r i z o n t a l I w a s a w a T h e o r y w o r k a n d i m m e r s e d h i m s e l f d a y a n d n i g h t i n t h e w o r k o f K o l y v a g i n a n d F l a c h . I f t h e i r " E u l e r S y s t e m " r e a l l y w o r k e d , W i l e s w o u l d h o p e f u l l y a c h i e v e t h e C l a s s N u m b e r r e s u l t a n d t h e S h i m u r a - T a n i y a m a C o n j e c t u r e w o u l d b e p r o v e d f o r s e m i s t a b l e e l l i p t i c c u r v e s —e n o u g h t o p r o v e F e r m a t ' s L a s t T h e o r e m .

T h i s w a s h a r d w o r k , h o w e v e r , a n d i t w a s o u t s i d e t h e I w a s a w a r e a l m w h i c h W i l e s k n e w s o w e l l . I n c r e a s i n g l y , W i l e s s t a r t e d t o f e e l t h e n e e d t o f i n d s o m e b o d y t o t a l k t o . H e w a n t e d s o m e o n e w h o c o u l d c h e c k h i s p r o g r e s s i n t h e s e u n c h a r t e d w a t e r s , b u t

s o m e o n e w h o w o u l d n o t r e v e a l a t h i n g t o a n y o n e e l s e .

A Good Friend

W i l e s f i n a l l y h a d t o m a k e a d e c i s i o n : s h o u l d h e c o n t i n u e t o k e e p e v e r y t h i n g s e c r e t a s h e h a d d o n e f o r s o l o n g , o r s h o u l d h e b r e a k d o w n a n d t a l k t o s o m e o n e w i t h g o o d k n o w l e d g e o f n u m b e r t h e o r y ? H e f i n a l l y d e c i d e d h e c o u l d p r o b a b l y n o t d o v e r y w e l l b y k e e p i n g s e c r e c y f o r e v e r . A s h e h i m s e l f s a i d , o n e

c o u l d w o r k o n a p r o b l e m f o r a n e n t i r e l i f e t i m e a n d n o t s e e a n y r e s u l t s . T h e n e e d t o c o m p a r e n o t e s w i t h a n o t h e r p e r s o n f i n a l l y o u t w e i g h e d t h a t i n t e n s e n e e d t o k e e p i t a l l t o h i m s e l f . B u t n o w t h e q u e s t i o n w a s : w h o ? W h o m c o u l d h e t r u s t t o k e e p h i s s e c r e t ?

I n J a n u a r y o f 1993, a f t e r s i x y e a r s o f w o r k i n g a l o n e , W i l e s m a d e h i s c o n t a c t . H e c a l l e d i n P r o f e s s o r N i c k K a t z , o n e o f h i s c o l l e a g u e s a t t h e P r i n c e t o n m a t h e m a t i c s d e p a r t m e n t . K a t z w a s a n e x p e r t o n m a n y o f t h e t h e o r i e s t h a t w e n t i n t o a t t e m p t s t o p r o v e t h e C l a s s N u m b e r F o r m u l a . B u t m o r e i m p o r t a n t l y , K a t z w a s c o m p l e t e l y t r u s t w o r t h y . H e w o u l d n e v e r r e v e a l w h a t A n d r e w W i l e s w a s u p t o . T h i s a s s e s s m e n t o n W i l e s ' p a r t t u r n e d o u t t o b e c o r r e c t . N i c k K a t z k e p t h i s m o u t h s h u t t h r o u g h o u t h i s w o r k w i t h W i l e s o v e r t h e m a n y r e m a i n i n g m o n t h s o f t h e p r o j e c t . T h e i r m u t u a l c o l l e a g u e s a t t h e t i g h t l y -k n i t m a t h e m a t i c a l c o m m u n i t y a t P r i n c e t o n n e v e r s u s p e c t e d a t h i n g , e v e n a f t e r w e e k s o f s e e i n g t h e t w o s p e n d h o u r s h u d d l e d t o g e t h e r o v e r c o f f e e a t t h e f a r e n d o f t h e C o m m o n s R o o m .

B u t A n d r e w W i l e s s t i l l w o r r i e d t h a t s o m e o n e m i g h t s u s p e c t w h a t h e w a s w o r k i n g o n . H e c o u l d n ' t t a k e a c h a n c e . S o h e c a m e u p w i t h a s c h e m e t o h i d e t h e f a c t t h a t h e w a s w o r k i n g v e r y i n t e n s e l y o n " s o m e t h i n g " w i t h N i c k K a t z . W i l e s w o u l d

o f f e r a n e w g r a d u a t e c o u r s e i n m a t h e m a t i c s f o r t h e s p r i n g o f 1993, a c o u r s e w h i c h N i c k K a t z w o u l d a t t e n d a s o n e o f t h e s t u d e n t s , a n d t h i s w o u l d a l l o w t h e t w o o f t h e m t o w o r k t o g e t h e r w i t h o u t o t h e r s s u s p e c t i n g w h a t t h e y w e r e d o i n g . O r a t l e a s t t h i s i s w h a t W i l e s h a s s a i d . T h e g r a d u a t e s t u d e n t s c o u l d n o t s u s p e c t t h a t b e h i n d t h e s e l e c t u r e s w a s a r o a d t o F e r m a t ' s L a s t T h e o r e m , a n d W i l e s w o u l d b e a b l e t o p i c k t h e i r b r a i n s f o r a n y p o s s i b l e h o l e s i n h i s t h e o r y , w i t h t h e h e l p o f h i s g o o d f r i e n d K a t z .

T h e c o u r s e w a s a n n o u n c e d . I t w a s c a l l e d " C a l c u l a t i o n s w i t h E l l i p t i c C u r v e s , " w h i c h w a s i n n o c e n t e n o u g h s o n o o n e c o u l d s u s p e c t a n y t h i n g . A n d a t t h e s t a r t o f t h e c o u r s e , P r o f e s s o r W i l e s s a i d t h a t t h e p u r p o s e o f t h e l e c t u r e s w a s t o s t u d y s o m e r e c e n t w o r k o f M a t t h i a s F l a c h o n t h e C l a s s N u m b e r F o r m u l a . T h e r e w a s n o m e n t i o n o f F e r m a t , n o m e n t i o n o f S h i m u r a o r T a n i y a m a , a n d n o o n e c o u l d s u s p e c t t h a t t h e C l a s s N u m b e r F o r m u l a t h e y w e r e g o i n g t o s t u d y w o u l d b e t h e k e y s t o n e t o p r o v i n g F e r m a t ' s L a s t T h e o r e m . A n d n o o n e h a d a n y i d e a t h a t t h e t r u e p u r p o s e o f t h e l e c t u r e s w a s n o t t o t e a c h g r a d u a t e s t u -d e n t s m a t h e m a t i c s b u t t o a l l o w W i l e s a n d K a t z t o w o r k t o g e t h e r o n t h i s p r o b l e m w i t h o u t s u s p i c i o n b y a n y o f t h e i r c o l l e a g u e s , w h i l e a t t h e s a m e t i m e g e t t i n g u n s u s p e c t i n g g r a d u a t e s t u d e n t s t o c h e c k t h e w o r k f o r t h e m .

B u t w i t h i n a f e w w e e k s a l l t h e g r a d u a t e s t u d e n t s d r i f t e d a w a y . T h e y c o u l d n ' t k e e p u p w i t h a c o u r s e t h a t w a s n ' t r e a l l y g o i n g a n y w h e r e . T h e o n l y " s t u d e n t " w h o s e e m e d t o k n o w a n y t h i n g a n d t o p a r t i c i p a t e i n c l a s s w a s t h e o t h e r m a t h p r o f e s s o r w h o w a s s i t t i n g t h e r e w i t h t h e m . S o a f t e r a w h i l e , N i c k K a t z w a s t h e o n l y o n e i n t h e a u d i e n c e . B u t W i l e s j u s t w e n t o n u s i n g t h e " c l a s s " t o w r i t e h i s l o n g p r o o f o f t h e

C l a s s N u m b e r T h e o r e m o n t h e b o a r d , c o n t i n u i n g t o t h e n e x t s t e p e v e r y c l a s s m e e t i n g , w i t h N i c k K a t z v e r i f y i n g e a c h s t e p .

T h e l e c t u r e s r e v e a l e d n o e r r o r s . I t s e e m e d t h a t t h e C l a s s N u m b e r F o r m u l a w a s w o r k i n g , a n d W i l e s w a s o n h i s w a y t o t h e s o l u t i o n o f t h e F e r m a t p r o b l e m . A n d s o i n t h e l a t e s p r i n g o f 1993, a s t h e c o u r s e w a s c o m i n g t o a n e n d , A n d r e w W i l e s w a s a l m o s t f i n i s h e d . S t i l l , h e w a s w r e s t l i n g w i t h j u s t o n e f i n a l o b s t a c l e . H e w a s a b l e t o p r o v e t h a t m o s t o f t h e e l l i p t i c c u r v e s

w e r e m o d u l a r , b u t a f e w o f t h e m r e m a i n e d u n p r o v a b l e . H e t h o u g h t h e c o u l d o v e r c o m e t h e s e d i f f i c u l t i e s , a n d h e w a s g e n e r a l l y o p t i m i s t i c . W i l e s f e l t i t w a s t i m e t o t a l k t o o n e m o r e p e r s o n , t o t r y t o g a i n a l i t t l e m o r e i n s i g h t i n t o t h i s l a s t d i f f i c u l t y f a c i n g h i m . S o h e c a l l e d i n a n o t h e r o n e o f h i s c o l l e a g u e s a t t h e P r i n c e t o n m a t h e m a t i c s d e p a r t m e n t , P r o f e s s o r P e t e r S a r n a k , a n d s w o r e h i m t o s e c r e c y a s w e l l . " 1 t h i n k I a m a b o u t t o p r o v e F e r m a t ' s L a s t T h e o r e m , " h e t o l d t h e s t u n n e d S a r n a k .

" T h i s w a s i n c r e d i b l e , " S a r n a k l a t e r r e c a l l e d . " 1 w a s f l a b b e r g a s t e d , e l a t e d , d i s t u r b e d — I m e a n . . . I r e m e m b e r f i n d i n g i t d i f f i c u l t t o s l e e p t h a t n i g h t . " S o n o w t h e r e w e r e t w o c o l l e a g u e s t r y i n g t o h e l p W i l e s f i n i s h h i s p r o o f . W h i l e n o b o d y s u s p e c t e d w h a t i t w a s t h a t t h e y w e r e d o i n g , p e o p l e w e r e n o t i c i n g s o m e t h i n g . A n d w h i l e h e m a i n t a i n e d t h a t n o o n e e v e r f o u n d a n y -t h i n g t h r o u g h h i m , S a r n a k l a t e r a d m i t t e d t h a t h e d r o p p e d " m a y b e a f e w h i n t s . . . "

The Last Piece oj the Puzzle

I n M a y , 1993, A n d r e w W i l e s w a s s i t t i n g a l o n e a t h i s d e s k . H e w a s g e t t i n g s o m e w h a t f r u s t r a t e d . I t s e e m e d t h a t t h o s e f e w e l l i p t i c c u r v e s t h a t g o t a w a y f r o m h i m w e r e n o t c o m i n g a n y n e a r e r . H e

s i m p l y c o u l d n ' t p r o v e t h a t t h e y w e r e m o d u l a r . A n d h e n e e d e d t h e m , t o o , i f h e w e r e t o p r o v e t h a t all ( s e m i s t a b l e ) e l l i p t i c c u r v e s w e r e m o d u l a r s o t h a t F e r m a t ' s L a s t T h e o r e m w o u l d f o l l o w . D o i n g i t f o r m o s t o f t h e s e m i s t a b l e e l l i p t i c c u r v e s w a s a g r e a t m a t h e m a t i c a l r e s u l t o n i t s o w n r i g h t , b u t n o t e n o u g h t o r e a c h h i s g o a l . To r e s t a l i t t l e h o r n h i s i n t e n s e s e a r c h l e a d i n g n o w h e r e , W i l e s p i c k e d u p a n o l d p a p e r o f t h e g r e a t m a s t e r , B a r r y M a z u r o f H a r v a r d U n i v e r s i t y . M a z u r h a d m a d e s o m e

g r o u n d b r e a k i n g d i s c o v e r i e s i n n u m b e r t h e o r y — r e s u l t s t h a t h a d i n s p i r e d m a n y o f t h e e x p e r t s i n t h e f i e l d , i n c l u d i n g R i b e t a n d F r e y , w h o s e w o r k p a v e d t h e w a y f o r W i l e s ' e f f o r t . M a z u r ' s p a p e r t h a t W i l e s w a s n o w r e r e a d i n g w a s a n e x t e n s i o n o f t h e t h e o r y o f i d e a l s , s t a r t i n g w i t h K u m m e r a n d D e d e k i n d , a n d c o n t i n u i n g w i t h y e t a t h i r d n i n e t e e n t h c e n t u r y m a t h e m a t i c i a n —F e r d i n a n d G o t t h o l d E i s e n s t e i n (1823-1852). A l t h o u g h h e d i e d y o u n g , E i s e n s t e i n m a d e g r e a t s t r i d e s i n n u m b e r t h e o r y . G a u s s , i n f a c t , i s q u o t e d a s h a v i n g s a i d : ' T h e r e h a v e b e e n o n l y t h r e e e p o c h - m a k i n g m a t h e m a t i c i a n s , A r c h i m e d e s , N e w t o n , a n d E i s e n s t e i n . "

M a z u r ' s p a p e r o n t h e E i s e n s t e i n I d e a l h a d o n e l i n e i n i t t h a t n o w c a u g h t W i l e s ' a t t e n t i o n . M a z u r w a s s a y i n g t h a t i t w a s p o s s i b l e t o switch

f r o m o n e s e t o f e l l i p t i c c u r v e s t o a n o t h e r . T h e s w i t c h h a d t o d o w i t h p r i m e n u m b e r s . W h a t M a z u r w a s s a y i n g w a s t h a t i f o n e w a s d e a l i n g w i t h e l l i p t i c c u r v e s t h a t w e r e b a s e d o n t h e p r i m e n u m b e r t h r e e , i t w a s p o s s i b l e t o t r a n s f o r m t h e m s o t h a t o n e c o u l d s t u d y t h e m u s i n g t h e p r i m e n u m b e r f i v e i n s t e a d . T h i s 3- t o -5 s w i t c h w a s e x a c t l y w h a t W i l e s w o u l d n e e d . H e w a s s t u c k w i t h n o t b e i n g a b l e t o p r o v e t h a t c e r t a i n c l a s s e s o f e l l i p t i c c u r v e s b a s e d o n t h e

p r i m e n u m b e r t h r e e w e r e m o d u l a r . A n d h e r e M a z u r w a s s a y i n g t h a t h e c o u l d s w i t c h t h e m t o c u r v e s b a s e d o n t h e p r i m e n u m b e r f i v e . B u t t h e s e c u r v e s b a s e d o n f i v e W i l e s h a d a l r e a d y p r o v e d w e r e m o d u l a r . S o t h e 3- t o -5 s w i t c h w a s t h e f i n a l t r i c k . I t t o o k t h e d i f f i c u l t e l l i p t i c c u r v e s b a s e d o n t h r e e , t r a n s f o r m e d t h e m i n t o b a s e f i v e a n d t h e s e w e r e k n o w n t o b e m o d u l a r . O n c e a g a i n , s o m e o t h e r m a t h e m a t i c i a n ' s b r i l l i a n t i d e a h e l p e d W i l e s o v e r c o m e a s e e m i n g l y i n s u r m o u n t a b l e h u r d l e . A n d r e w W i l e s w a s f i n a l l y d o n e .

H i s t i m i n g w a s p e r f e c t , t o o . I n t h e n e x t m o n t h , J u n e , h i s

f o r m e r a d v i s e r J o h n C o a t e s w o u l d b e h o s t i n g a n u m b e r t h e o r y c o n f e r e n c e i n C a m b r i d g e . A l l t h e b i g n a m e s i n n u m b e r t h e o r y w o u l d b e t h e r e . A n d C a m b r i d g e w a s W i l e s ' o l d h o m e t o w n a n d w h e r e h e w e n t t o g r a d u a t e s c h o o l . W o u l d n ' t i t b e p e r f e c t t o p r e s e n t h i s p r o o f o f F e r m a t ' s L a s t T h e o r e m t h e r e ? W i l e s w a s n o w r a c i n g a g a i n s t t h e c l o c k . H e h a d t o p u t t o g e t h e r h i s e n t i r e p r o o f t h a t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s t r u e f o r s e m i s t a b l e e l l i p t i c c u r v e s . T h i s m e a n t t h a t t h e F r e y c u r v e c o u l d n o t e x i s t . A n d i f t h e F r e y c u r v e c o u l d n o t " e x i s t , t h a t m e a n t t h a t s o l u t i o n s t o F e r m a t ' s e q u a t i o n c o u l d n o t e x i s t f o r n>2 a n d t h e r e f o r e F e r m a t ' s L a s t T h e o r e m w a s p r o v e n . T h e w r i t e - u p t o o k A n d r e w W i l e s 200 p a g e s . H e w a s d o n e j u s t i n t i m e t o c a t c h h i s p l a n e f o r E n g l a n d . A n d a t t h e e n d o f h i s l a s t l e c t u r e t h e r e , h e e m e r g e d v i c t o r i o u s t o t h e s w e e p i n g a p p l a u s e , t h e f l a s h i n g c a m e r a s , a n d t h e r e p o r t e r s .

The Aftermath

N o w i t w a s t i m e f o r t h e p e e r - r e v i e w . U s u a l l y , a m a t h e m a t i c a l r e s u l t — o r a n y a c a d e m i c f i n d i n g , f o r t h a t m a t t e r — i s s u b m i t t e d t o a " r e f e r e e d j o u r n a l . " S u c h r e f e r e e d j o u r n a l s a r e t h e s t a n d a r d v e h i c l e f o r s c h o l a r s t o s u b m i t t h e i r w o r k f o r p o s s i b l e p u b l i c a t i o n . T h e

j o u r n a l ' s c h a r g e i s t h e n t o s e n d t h e p r o p o s e d p a p e r t o o t h e r e x p e r t s i n t h e a p p r o p r i a t e a r e a o f s t u d y w h o r e v i e w t h e p a p e r ' s c o n t e n t a n d d e t e r m i n e w h e t h e r i t i s c o r r e c t , a n d w h e t h e r t h e p a p e r m a k e s a c o n t r i b u t i o n w o r t h y o f p u b l i c a t i o n . R e f e r e e d j o u r n a l p u b l i c a t i o n s a r e t h e b r e a d a n d b u t t e r o f a c a d e m i a . T e n u r e a n d p r o m o t i o n , a n d o f t e n s a l a r y l e v e l s a n d r a i s e s , a r e a l l d e p e n d e n t o n a r e s e a r c h e r ' s o u t p u t o f r e f e r e e d j o u r n a l a r t i c l e s .

B u t A n d r e w W i l e s c h o s e a d i f f e r e n t a p p r o a c h . I n s t e a d o f s u b m i t t i n g h i s p r o o f t o a p r o f e s s i o n a l m a t h e m a t i c s j o u r n a l — a s a l m o s t a n y o n e e l s e w o u l d h a v e d o n e — h e p r e s e n t e d i t a t a c o n f e r e n c e . T h e r e a s o n w a s p r o b a b l y t w o f o l d . T h r o u g h o u t t h e y e a r s o f w o r k o n t h e p r o o f , W i l e s w a s o b s e s s e d w i t h s e c r e c y . I f h e s u b m i t t e d t h e p r o o f t o a j o u r n a l , t h e p r o o f w o u l d h a v e b e e n s e n t t o a n u m b e r o f r e f e r e e s c h o s e n b y t h e j o u r n a l a n d o n e o f t h e m , o r t h e e d i t o r s , m i g h t h a v e s a i d s o m e t h i n g t o t h e w o r l d a t l a r g e . W i l e s p r o b a b l y w a s a l s o w o r r i e d t h a t s o m e o n e w h o r e a d t h e p r o p o s e d p r o o f m i g h t s o m e h o w s t e a l i t a n d s e n d i t o u t u n d e r h i s o r h e r o w n n a m e . T h i s , u n f o r t u n a t e l y , d o e s h a p p e n i n a c a d e m i a . T h e o t h e r r e a s o n , l i n k e d w i t h t h e f i r s t , w a s t h a t W i l e s w a n t e d t o m a i n t a i n t h e b u i l d u p o f s u s p e n s e a s h e p r e s e n t e d h i s p r o o f a t C a m b r i d g e .

B u t e v e n s o , h a v i n g p r e s e n t e d t h e r e s u l t s a t a c o n f e r e n c e , t h e w o r k w o u l d s t i l l h a v e t o b e r e f e r e e d . T h e s t e p s w o u l d s t i l l h a v e t o b e p e e r - r e v i e w e d , t h a t i s , o t h e r e x p e r t s i n n u m b e r t h e o r y w o u l d h a v e t o g o t h r o u g h W i l e s ' p r o o f , l i n e b y l i n e , t o a s c e r t a i n t h a t h e h a d i n d e e d e s t a b l i s h e d w h a t h e s e t o u t t o p r o v e .

The Deep Gulf Materializes

W i l e s ' 200- p a g e p a p e r w a s

s e n t t o a n u m b e r o f l e a d i n g e x p e r t s i n n u m b e r t h e o r y . S o m e o f t h e m q u i c k l y e x p r e s s e d c o n c e r n s , b u t g e n e r a l l y m a t h e m a t i c i a n s t h o u g h t t h e p r o o f w a s p r o b a b l y c o r r e c t . T h e y w o u l d w a i t t o h e a r t h e e x p e r t s ' v e r d i c t , h o w e v e r . " O h y e s ! " s a i d K e n R i b e t w h e n I a s k e d h i m i f h e b e l i e v e d W i l e s ' p r o o f . " I w a s u n a b l e t o s e e w h a t s o m e p e o p l e w e r e s a y -i n g s o o n a f t e r t h e y r e a d t h e p r o o f — n a m e l y t h a t t h e r e w a s n o E u l e r S y s t e m h e r e . "

O n e o f t h e e x p e r t s c h o s e n t o g o o v e r W i l e s ' p r o o f w a s h i s P r i n c e t o n f r i e n d , N i c k K a t z . P r o f e s s o r K a t z s p e n t t w o w h o l e m o n t h s , J u l y a n d A u g u s t o f 1993, d o i n g n o t h i n g b u t s t u d y i n g t h e e n t i r e p r o o f . E v e r y d a y , h e w o u l d s i t a t h i s d e s k a n d s l o w l y r e a d e v e r y l i n e , e v e r y m a t h e m a t i c a l s y m b o l , e v e r y l o g i c a l i m p l i c a t i o n , t o m a k e s u r e t h a t i t m a d e p e r f e c t s e n s e a n d t h a t i t w o u l d i n d e e d b e a c c e p t a b l e t o a n y m a t h e m a t i c i a n w h o m i g h t r e a d t h e p r o o f . O n c e o r t w i c e a d a y , K a t z w o u l d s e n d a n e - m a i l m e s s a g e t o A n d r e w W i l e s , w h o s t a y e d a w a y f r o m P r i n c e t o n t h a t s u m m e r , a s k i n g h i m : " W h a t d o y o u m e a n i n t h i s l i n e o n t h i s p a g e ? " o r " I d o n ' t s e e h o w t h i s i m p l i c a t i o n f o l l o w s f r o m t h e o n e a b o v e , " e t c . I n r e s p o n s e , W i l e s w o u l d e - m a i l b a c k , a n d i f t h e p r o b l e m r e q u i r e d m o r e d e t a i l s , h e w o u l d f a x t h e a n s w e r t o K a t z .

O n e d a y , w h e n K a t z w a s a b o u t t w o t h i r d s o f t h e w a y t h r o u g h W i l e s ' l o n g m a n u s c r i p t , h e c a m e a c r o s s a p r o b l e m . I t s e e m e d i n n o c e n t e n o u g h a t f i r s t , o n e o f m a n y W i l e s h a d a n s w e r e d e a r l i e r t o K a t z ' s c o m p l e t e s a t i s f a c t i o n . B u t n o t t h i s t i m e . I n r e s p o n s e t o K a t z ' s q u e s t i o n s , W i l e s e -m a i l e d b a c k a n a n s w e r . B u t K a t z h a d t o e - m a i l b a c k : " I s t i l l d o n ' t u n d e r s t a n d i t , A n d r e w . " S o t h i s t i m e W i l e s s e n t a f a x t r y i n g t o m a k e t h e l o g i c a l c o n n e c t i o n .

A g a i n K a t z w a s n o t s a t i s f i e d . S o m e t h i n g w a s s i m p l y n o t r i g h t . T h i s w a s s u p p o s e d t o b e e x a c t l y o n e o f t h e a r g u m e n t s t h a t W i l e s a n d K a t z w e n t o v e r c a r e f u l l y i n t h e s p r i n g w h e n W i l e s w a s t e a c h i n g h i s " c o u r s e . " A n y d i f f i c u l t i e s s h o u l d h a v e a l r e a d y b e e n i r o n e d o u t . B u t a p p a r e n t l y t h e h o l e i n W i l e s ' l o g i c e l u d e d t h e m b o t h . P o s s i b l y i f t h e g r a d u a t e s t u -d e n t s h a d s t a y e d o n , o n e o f t h e m m i g h t h a v e a l e r t e d t h e t w o t o t h e p r o b l e m .

B y t h e t i m e K a t z f o u n d t h e e r r o r , o t h e r m a t h e m a t i c i a n s t h r o u g h o u t t h e w o r l d w e r e a w a r e o f t h e e x a c t s a m e p r o b l e m w i t h W i l e s ' p r o o f . T h e r e s i m p l y w a s n o E u l e r S y s t e m h e r e , a n d t h e r e w a s n o t h i n g d o i n g . A n d w i t h o u t t h e E u l e r S y s t e m — s u p -p o s e d l y a g e n e r a l i z a t i o n o f t h e e a r l i e r w o r k o f F l a c h a n d K o l y - v a g i n — t h e r e w a s n o C l a s s N u m b e r F o r m u l a . W i t h o u t t h e C l a s s N u m b e r F o r m u l a i t w a s i m p o s s i b l e t o " c o u n t " t h e G a l o i s r e p r e s e n t a t i o n s o f t h e e l l i p t i c c u r v e s a g a i n s t t h e m o d u l a r f o r m s , a n d S h i m u r a -T a n i y a m a w a s n o t e s t a b l i s h e d . A n d w i t h o u t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e p r o v e d a s c o r r e c t , t h e r e w a s n o p r o o f o f F e r m a t ' s L a s t T h e o r e m . I n s h o r t , t h e h o l e i n t h e E u l e r S y s t e m m a d e e v e r y t h i n g c o l l a p s e l i k e a h o u s e o f c a r d s .

The Agony

A n d r e w W i l e s r e t u r n e d t o P r i n c e t o n i n t h e f a l l o f 1993.

H e w a s e m b a r r a s s e d , h e w a s u p s e t , h e w a s a n g r y , f r u s t r a t e d , h u m i l i a t e d . H e h a d p r o m i s e d t h e w o r l d a p r o o f o f F e r m a t ' s L a s t T h e o r e m — b u t h e c o u l d n ' t d e l i v e r . I n m a t h e m a t i c s , a s i n a l m o s t a n y o t h e r f i e l d , t h e r e a r e n o r e a l " s e c o n d p r i z e s " o r " a l s o r a n " a w a r d s . T h e c r e s t f a l l e n W i l e s w a s b a c k i n h i s a t t i c t r y i n g t o f i x t h e p r o o f . " A t t h i s p o i n t , h e w a s h i d i n g a s e c r e t f r o m

t h e w o r l d , " r e c a l l e d N i c k K a t z , " a n d I t h i n k h e m u s t h a v e f e l t p r e t t y u n c o m f o r t a b l e a b o u t i t . " O t h e r c o l l e a g u e s h a d t r i e d t o h e l p W i l e s , i n c l u d i n g h i s f o r m e r s t u d e n t R i c h a r d T a y l o r w h o w a s t e a c h i n g a t C a m b r i d g e b u t j o i n e d W i l e s a t P r i n c e t o n t o h e l p h i m t r y t o f i x t h e p r o o f .

" T h e f i r s t s e v e n y e a r s , w o r k i n g a l l a l o n e , I e n j o y e d e v e r y m i n u t e o f i t , " W i l e s r e c a l l e d , " n o m a t t e r h o w h a r d o r s e e m i n g l y i m p o s s i b l e a h u r d l e I f a c e d . B u t n o w , d o i n g m a t h e m a t i c s

i n t h i s o v e r - e x p o s e d w a y w a s c e r t a i n l y n o t m y s t y l e . I h a v e n o w i s h t o e v e r r e p e a t t h i s e x p e r i e n c e . " A n d t h e b a d e x p e r i e n c e l a s t e d a n d l a s t e d . R i c h a r d T a y l o r , h i s s a b b a t i c a l l e a v e o v e r , r e t u r n e d t o C a m b r i d g e a n d s t i l l W i l e s s a w n o e n d i n s i g h t . H i s c o l l e a g u e s l o o k e d a t h i m w i t h a m i x t u r e o f a n t i c i p a t i o n , h o p e , a n d p i t y , a n d h i s s u f f e r i n g w a s c l e a r t o e v e r y o n e a r o u n d h i m . P e o p l e w a n t e d t o k n o w . T h e y w a n t e d t o h e a r g o o d n e w s , b u t n o n e o f h i s c o l l e a g u e s d a r e d a s k h i m h o w h e w a s d o i n g w i t h t h e p r o o f . O u t s i d e h i s d e p a r t m e n t , t h e r e s t o f t h e w o r l d w a s c u r i o u s , t o o . S o m e t i m e i n t h e n i g h t o f D e c e m b e r 4, 1993,

A n d r e w W i l e s p o s t e d a n e -m a i l m e s s a g e t o t h e c o m p u t e r n e w s g r o u p S c i . m a t h , t o w h i c h s e v e r a l n u m b e r t h e o r i s t s a n d o t h e r m a t h e m a t i c i a n s b e l o n g e d :

In view of the speculation on the status of my

work on the Taniyama-Shimura conjecture and

Fermat's Last Theorem 1 will give a brief account

of the situation. During the review process a

number of problems emerged, most of which have

been resolved, but one in particular 1 have not yet

settled... I believe that 1 will be able to finish this

in the near future using the ideas explained in my

Cambridge lectures. The fact that a lot of work

remains to be done on the manuscript makes it

unsuitable for release as a preprint. In my course

in Princeton beginning in February I will give a full

account of this work.

Andrew Wiles

The Post-Mortem

B u t A n d r e w W i l e s w a s

p r e m a t u r e l y o p t i m i s t i c . A n d w h a t e v e r c o u r s e h e m a y h a v e p l a n n e d t o o f f e r a t P r i n c e t o n w o u l d n o t

y i e l d a n y s o l u t i o n . W h e n m o r e t h a n a y e a r p a s s e d s i n c e h i s s h o r t - l i v e d t r i u m p h i n C a m b r i d g e , A n d r e w W i l e s w a s a b o u t t o g i v e u p a l l h o p e a n d t o f o r g e t h i s c r i p p l e d p r o o f .

O n M o n d a y m o r n i n g , S e p t e m b e r 19, 1994, W i l e s w a s s i t t i n g a t h i s d e s k a t P r i n c e t o n U n i v e r s i t y , p i l e s o f p a p e r s t r e w n a l l a r o u n d h i m . H e d e c i d e d h e w o u l d t a k e o n e l a s t l o o k a t h i s p r o o f b e f o r e c h u c k i n g i t a l l a n d a b a n d o n i n g a l l h o p e t o p r o v e F e r m a t ' s L a s t T h e o r e m . H e w a n t e d t o s e e e x a c t l y w h a t i t w a s t h a t w a s p r e v e n t i n g h i m f r o m c o n s t r u c t i n g t h e E u l e r S y s t e m . H e w a n t e d t o k n o w —j u s t f o r h i s o w n s a t i s f a c t i o n— w h y h e h a d f a i l e d . W h y w a s t h e r e n o E u l e r S y s t e m ? — h e w a n t e d t o b e a b l e t o p i n p o i n t p r e c i s e l y w h i c h t e c h n i c a l f a c t w a s m a k i n g t h e w h o l e t h i n g f a i l . I f h e w a s g o i n g t o g i v e u p , h e f e l t , t h e n a t l e a s t h e w a s o w e d a n a n s w e r t o w h y h e h a d b e e n w r o n g .

W i l e s s t u d i e d t h e p a p e r s i n f r o n t o f h i m , c o n c e n t r a t i n g v e r y h a r d f o r a b o u t t w e n t y m i n u t e s . A n d t h e n h e s a w e x a c t l y w h y h e w a s u n a b l e t o m a k e t h e s y s t e m w o r k . F i n a l l y , h e u n d e r s t o o d w h a t w a s w r o n g . " I t w a s t h e m o s t i m p o r t a n t m o m e n t i n m y e n t i r e w o r k i n g l i f e , " h e l a t e r d e s c r i b e d t h e f e e l i n g . " S u d d e n l y , t o t a l l y u n e x p e c t e d l y , I h a d t h i s i n c r e d i b l e r e v e l a t i o n .

N o t h i n g I ' l l e v e r d o a g a i n w i l l . . . " a t t h a t m o m e n t t e a r s w e l l e d u p a n d W i l e s w a s c h o k i n g w i t h e m o t i o n . W h a t W i l e s r e a l i z e d a t t h a t f a t e f u l m o m e n t w a s " s o i n d e s c r i b a b l y b e a u t i f u l , i t w a s s o s i m p l e a n d s o e l e g a n t . . . a n d I j u s t s t a r e d i n d i s b e l i e f . " W i l e s r e a l i z e d t h a t e x a c t l y w h a t w a s m a k i n g t h e E u l e r S y s t e m f a i l i s w h a t w o u l d m a k e t h e H o r i z o n t a l I w a s a w a T h e o r y a p p r o a c h h e h a d a b a n d o n e d t h r e e y e a r s e a r l i e r work. W i l e s s t a r e d a t h i s p a p e r f o r a l o n g t i m e . H e m u s t b e d r e a m i n g , h e t h o u g h t , t h i s w a s j u s t t o o g o o d t o b e t r u e . B u t l a t e r h e s a i d i t w a s s i m p l y t o o g o o d t o b e

false. T h e d i s c o v e r y w a s s o p o w e r f u l , s o b e a u t i f u l , t h a t i t bad t o b e t r u e .

W i l e s w a l k e d a r o u n d t h e d e p a r t m e n t f o r s e v e r a l h o u r s . H e d i d n ' t k n o w w h e t h e r h e w a s a w a k e o r d r e a m i n g . E v e r y o n c e i n a w h i l e , h e w o u l d r e t u r n t o h i s d e s k t o s e e i f h i s f a n t a s t i c f i n d i n g w a s s t i l l t h e r e — a n d i t w a s . H e w e n t h o m e . H e h a d t o s l e e p o n i t — m a y b e i n t h e m o r n i n g h e w o u l d f i n d s o m e f l a w i n t h i s n e w a r g u m e n t . A y e a r o f p r e s s u r e f r o m t h e e n t i r e w o r l d , a y e a r o f o n e f r u s t r a t e d a t t e m p t a f t e r a n o t h e r h a d s h a k e n W i l e s ' c o n f i d e n c e . H e c a m e b a c k t o h i s d e s k i n t h e m o r n i n g , a n d t h e i n c r e d i b l e g e m h e h a d f o u n d t h e d a y b e f o r e w a s s t i l l t h e r e , w a i t i n g f o r h i m .

W i l e s w r o t e u p h i s p r o o f u s i n g t h e c o r r e c t e d H o r i z o n t a l I w a s a w a T h e o r y a p p r o a c h . F i n a l l y , e v e r y t h i n g f e l l p e r f e c t l y i n t o p l a c e . T h e a p p r o a c h h e h a d u s e d t h r e e y e a r s e a r l i e r w a s t h e c o r r e c t o n e . A n d t h a t k n o w l e d g e c a m e t o h i m f r o m t h e f a i l i n g o f t h e F l a c h a n d K o l y v a g i n r o u t e h e h a d c h o s e n i n m i d s t r e a m . T h e m a n u s c r i p t w a s r e a d y t o b e s h i p p e d o u t . E l a t e d , A n d r e w

W i l e s l o g g e d i n t o h i s c o m p u t e r a c c o u n t . H e s e n t e -m a i l m e s s a g e s a c r o s s t h e I n t e r n e t t o a s c o r e o f m a t h e m a t i c i a n s a r o u n d t h e w o r l d : " E x p e c t a F e d e r a l E x p r e s s p a c k a g e i n t h e n e x t f e w d a y s , " t h e m e s s a g e s r e a d .

A s h e h a d p r o m i s e d h i s f r i e n d R i c h a r d T a y l o r , w h o h a d c o m e f r o m E n g l a n d e s p e c i a l l y t o h e l p h i m c o r r e c t h i s p r o o f , t h e n e w p a p e r c o r r e c t i n g t h e I w a s a w a t h e o r y b o r e b o t h o f t h e i r n a m e s e v e n t h o u g h W i l e s o b t a i n e d t h e a c t u a l r e s u l t a f t e r T a y l o r ' s d e p a r t u r e . W i t h i n t h e n e x t f e w w e e k s , t h e m a t h e m a t i c i a n s w h o r e c e i v e d W i l e s ' c o r r e c t i o n t o h i s C a m b r i d g e p a p e r s w e n t o v e r a l l t h e d e t a i l s . T h e y c o u l d f i n d n o t h i n g w r o n g .

I 33

W i l e s n o w u s e d t h e c o n v e n t i o n a l a p p r o a c h t o t h e p r e s e n t a t i o n o f m a t h e m a t i c a l r e s u l t s . I n s t e a d o f d o i n g w h a t h e h a d d o n e i n C a m b r i d g e a y e a r a n d a h a l f e a r l i e r , h e s e n t t h e p a p e r s t o a p r o f e s s i o n a l j o u r n a l , t h e Annals oj Mathematics, w h e r e t h e y c o u l d b e p e e r - r e v i e w e d b y o t h e r m a t h e m a t i c i a n s . T h e r e v i e w p r o c e s s t o o k a f e w m o n t h s , b u t n o f l a w s w e r e f o u n d t h i s t i m e . T h e M a y , 1995, i s s u e o f t h e j o u r n a l c o n t a i n e d W i l e s ' o r i g i n a l C a m b r i d g e p a p e r a n d t h e c o r r e c t i o n b y T a y l o r a n d W i l e s . F e r m a t ' s L a s t T h e o r e m w a s f i n a l l y l a i d t o r e s t .

Did Fermat Possess the Prooj?

A n d r e w W i l e s d e s c r i b e d h i s p r o o f a s " a t w e n t i e t h c e n t u r y p r o o f . " I n d e e d , W i l e s u s e d t h e w o r k o f m a n y t w e n t i e t h - c e n t u r y m a t h e m a t i c i a n s . H e a l s o u s e d t h e w o r k o f e a r l i e r m a t h e m a t i c i a n s . A l l t h e m y r i a d e l e m e n t s o f W i l e s ' c o n s t r u c t i o n s c a m e f r o m t h e w o r k o f o t h e r s , m a n y o t h e r s . S o t h e p r o o f o f F e r m a t ' s L a s t T h e o r e m w a s r e a l l y t h e a c h i e v e m e n t o f a l a r g e n u m b e r o f m a t h e m a t i c i a n s l i v i n g i n t h e t w e n t i e t h c e n t u r y — a n d a l l t h e p r e c e d i n g o n e s t o t h e t i m e o f F e r m a t h i m s e l f . A c c o r d i n g t o W i l e s , F e r m a t c o u l d n o t p o s s i b l y h a v e h a d t h i s p r o o f i n m i n d w h e n h i s w r o t e h i s f a m o u s n o t e i n t h e m a r g i n . T h i s m u c h i s t r u e , o f c o u r s e , b e c a u s e t h e S h i m u r a -

T a n i y a m a c o n j e c t u r e d i d n o t e x i s t u n t i l t h e t w e n t i e t h c e n t u r y . B u t c o u l d F e r m a t h a v e h a d a n o t h e r p r o o f i n m i n d ?

T h e a n s w e r i s p r o b a b l y n o t . B u t t h i s i s n o t a c e r t a i n t y . We w i l l n e v e r k n o w . O n t h e o t h e r h a n d , F e r m a t l i v e d a n o t h e r 28 y e a r s a f t e r h e w r o t e h i s t h e o r e m i n t h e m a r g i n . A n d h e n e v e r s a i d a n y t h i n g m o r e a b o u t i t . P o s s i b l y h e k n e w h e c o u l d n ' t p r o v e t h e t h e o r e m . O r h e m a y h a v e e r r o n e o u s l y t h o u g h t t h a t

h i s m e t h o d o f i n f i n i t e d e s c e n t u s e d i n p r o v i n g t h e s i m p l e c a s e « = 3 c o u l d a p p l y t o a g e n e r a l s o l u t i o n . O r m a y b e h e s i m p l y f o r g o t a b o u t t h e t h e o r e m a n d w e n t o n t o d o o t h e r t h i n g s .

P r o v i n g t h e t h e o r e m t h e w a y i t w a s f i n a l l y d o n e i n t h e 1990s r e q u i r e d a l o t m o r e m a t h e m a t i c s t h a n F e r m a t h i m s e l f c o u l d h a v e k n o w n . T h e p r o f o u n d n a t u r e o f t h e t h e o r e m i s t h a t n o t o n l y d o e s i t s h i s t o r y s p a n t h e l e n g t h o f h u m a n c i v i l i z a t i o n , b u t t h e f i n a l s o l u t i o n o f t h e p r o b l e m c a m e a b o u t b y h a r n e s s i n g — a n d i n a s e n s e u n i f y i n g — t h e e n t i r e b r e a d t h o f m a t h e m a t i c s . I t w a s t h i s u n i f i c a t i o n o f s e e m i n g l y d i s p a r a t e a r e a s o f m a t h e m a t i c s t h a t f i n a l l y n a i l e d t h e t h e o r e m . A n d d e s p i t e t h e f a c t t h a t A n d r e w W i l e s w a s t h e p e r s o n w h o d i d t h e i m p o r t a n t f i n a l w o r k o n t h e t h e o r e m b y p r o v i n g a f o r m o f t h e S h i m u r a -T a n i y a m a c o n j e c t u r e n e e d e d t o p r o v e F e r m a t ' s t h e o r e m , t h e e n t i r e e n t e r p r i s e w a s t h e w o r k o f m a n y p e o p l e . A n d i t i s a l l t h e i r c o n t r i b u -t i o n s , t a k e n t o g e t h e r , w h i c h b r o u g h t a b o u t t h e f i n a l s o l u t i o n . W i t h o u t t h e w o r k o f E r n s t K u m m e r t h e r e w o u l d h a v e b e e n n o t h e o r y o f i d e a l s , a n d w i t h o u t i d e a l s t h e w o r k o f B a r r y M a z u r w o u l d n o t h a v e e x i s t e d . W i t h o u t M a z u r t h e r e w o u l d h a v e b e e n n o c o n j e c t u r e b y F r e y , a n d w i t h o u t t h e c r u c i a l c o n j e c t u r e a n d i t s s y n t h e s i s

b y S e r r e t h e r e w o u l d n o t h a v e b e e n t h e p r o o f b y R i b e t t h a t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w o u l d e s t a b l i s h F e r m a t ' s L a s t T h e o r e m . A n d i t s e e m s t h a t n o p r o o f o f F e r m a t ' s L a s t T h e o r e m w o u l d b e p o s s i b l e w i t h o u t t h a t c o n j e c t u r e p u t f o r w a r d b y Y u t a k a T a n i y a m a a t T o k y o -N i k k o i n 1955 a n d t h e n r e f i n e d a n d m a d e s p e c i f i c b y G o r o S h i m u r a . O r w o u l d i t ?

F e r m a t , o f c o u r s e , c o u l d n o t h a v e f o r m u l a t e d s u c h a n o v e r a r c h i n g c o n j e c t u r e t h a t w o u l d u n i f y t w o v e r y d i f f e r e n t b r a n c h e s o f m a t h e m a t i c s . O r c o u l d h e h a v e d o n e s o ? N o t h i n g

i s c e r t a i n . We o n l y k n o w t h a t t h e t h e o r e m h a s f i n a l l y b e e n e s t a b l i s h e d a n d t h a t t h e p r o o f h a s b e e n c h e c k e d a n d v e r i f i e d t o i t s f i n e s t d e t a i l s b y s c o r e s o f m a t h e m a t i c i a n s t h r o u g h o u t t h e w o r l d . B u t j u s t b e c a u s e a p r o o f e x i s t s a n d i t i s a v e r y c o m p l i c a t e d , a d v a n c e d o n e d o e s n o t m e a n t h a t a s i m p l e r p r o o f i s n o t p o s s i b l e . R i b e t , i n f a c t , p o i n t s o u t i n o n e o f h i s p a p e r s a d i r e c t i o n w h e r e a p r o o f o f F e r m a t ' s t h e o r e m m i g h t b e p o s s i b l e w i t h o u t a p r o o f o f t h e S h i m u r a -T a n i y a m a c o n j e c t u r e . A n d p e r h a p s F e r m a t d i d k n o w a l o t o f p o w e r f u l " m o d e r n " m a t h e m a t i c s , n o w l o s t ( a c t u a l l y , t h e c o p y o f B a c h e t ' s D i o p h a n t u s i n w h i c h h e s u p p o s e d l y w r o t e h i s m a r g i n - s t a t e m e n t h a s n e v e r b e e n f o u n d ) . S o w h e t h e r o r n o t F e r m a t d i d p o s s e s s a " t r u l y m a r v e l o u s p r o o f " o f h i s t h e o r e m , o n e t h a t c o u l d n o t f i t i n t h e m a r g i n o f h i s b o o k , w i l l f o r e v e r r e m a i n h i s s e c r e t .

Gerd Faltings. He had a

totally different approach

to Fermat's Last Theorem.

When Wiles failed in his

first attempt in 1993,

many feared that Faltings

would now beat him to the

true proof.

Andrew Wiles at the crucial

moment of his third lecture

at Cambridge, June 1993,

when it was clear to

everyone Fermat was just

around the corner.

Ken Ribet at the famous

cafe where he finished the

proof that Shimura-

Taniyama would imply that

Fermat's Last Theorem had

to be true.

Endnotes

1. E.T. Bell, Men oj Mathematics, New York: Simon and Schuster, 1937, p. 56 .

I . Barry Mazur, "Number Theory as Gadfly," American Mathematical

Monthly, Vol. 98 , 1991, p. 593.

3 . Plimpton 322 and its implications about the advanced level of

Babylonian mathematics were brought to the attention of the scientific com-

munity by Otto Neugebauer in 1934. An account in English can be found in

his book The Exact Sciences in Antiquity (Princeton University Press, 1957 ) .

23.Actually, Cantor went much farther. He hypothesized.that the order of

infinity of the irrational numbers immediately follows that of the rationals.

That is, he believed that there is no order of infinity that is both higher

than that of the rational numbers and lower than that of the irrational

numbers. This statement became known as the Continuum Hypothesis,

and the work of Kurt Godel and Paul Cohen in the twentieth century

established that it is impossible to prove this hypothesis within the rest of

mathematics. The Continuum Hypothesis stands alone (with some

equivalent restatements) opposite the rest of mathematics, their

respective truths independent of each other. This remains one of the most

bizarre truths in the foundations of mathematics.

24.D. Wells, Curious and Interesting Numbers, London: Penguin Books, 1987,

p. 81 .

25.C. Boyer, A History of Mathematics, New York: Wiley, 1968 , p. 9 .

26.Reprinted in B. Mazur, op. cit.

27.Ian Stewart, Nature's Numbers, New York: Basic Books, 1995, p. 140.

9 . Michael Mahoney, The Mathematical Career oj Pierre de Fermat, 2d ed.,

Princeton University Press, 1994, p. 4 .

10 . Harold M. Edwards, Fermat's Last Theorem, New York: Springer-Verlag,

1977, pp. 61-73 .

I I . Much of what is publicly known about the secret society comes from

Paul R. Halmos, "Nicolas Bourbaki," Scientific American, 196, May 1957, pp.

88 -97 .

12 . Andre Weil, Oeuvres, Vols. Mil, Paris: Springer-Verlag, 1979.

13 . Adapted from Kenneth A. Ribet and Brian Hayes, "Fermat's Last

Theorem and Modern Arithmetic," American Scientist, Vol. 82 , March-April

1994, pp. 144-156 .

28.A good introduction to this topic is the book by Joseph H . Silverman and

John Tate, Rational Points on Elliptic Curves, New York: Springer-Verlag,

1992.

29.Most of the information on Yutaka Taniyama's life is from Goro Shimura,

"Yutaka Taniyama and His Time: Very Personal Recollections," Bulletin oj

the London Mathematical Society, Vol. 21 , 1989, pp. 184-196 .

30.Reprinted in the Japanese journal Sugaku, May 1956, pp. 227 -231 .

31.Shimura thus stated to Serre his actual conjecture, sharing it for the

first time, and implicitly trusting that Serre would acknowledge him as its

originator.

32.Andre Weil, Oeuvres, op. cit., Vol. Ill, p. 450.

33.Andre Weil, "Liber die Bestimmung Dirichletscher Reihen durch

Funktionalgleichungen," Math. Annalen, Vol. 168 , 1967, pp. 165-172.

34.Weil's letter to Lang, along with much of the chronology of events

described here, including private conversations and letters, are

reproduced in Serge Lang, "Some History of the Shimura-Taniyama

Conjecture," Notices oj the American Mathematical Society, November 1995,

pp. 1301-1307. It is to Lang's credit that his article and the "Taniyama-

Shimura File" he has been circulating among mathematicians for ten

years now are finally helping bring Goro Shimura the recognition he

rightly deserves.

35.Jean-Pierre Serre, "Lettre a J.-F. Mestre," reprinted in Current Trends in

Arithmetical Algebraic Geometry, Providence: American Mathematical

Society, 1987, pp. 263 -268 .

22 . Barry Mazur, "Modular Curves and the Eisenstein Ideal," Paris,

France: The Mathematical Publications ojI.H.E.S., Vol. 47 , 1977, pp. 33-186.

36.Barry Mazur, op. cit.

37.The first and more important of the two papers, Andrew Wiles, "Modular

Elliptic Curves and Fermat's Last Theorem," Annals of Mathematics, Vol.

142, 1995, pp. 443-551, begins with Fermat's actual margin-

statement of his theorem in Latin: Cubum autem in duos cubos, aut

quadratocjuadratum in duos quadratoquadratos, et generaliter nullam in infinitum

ultra quadratum potestatem in duos ejus-dem nominis fas est dividere: cujus rei

demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Pierre de Fermat. The journal sold out even before the publication date, and

for the first time imposed a charge of $14 per individual issue.

Author's Note

I n p r e p a r i n g t h i s b o o k I d r e w m u c h o f t h e h i s t o r i c a l b a c k -g r o u n d f r o m a v a r i e t y o f s o u r c e s . M y f a v o r i t e , a n d t h e m o s t c o m p l e t e a n d o r i g i n a l s o u r c e , i s E . T . B e l l ' s b o o k , Men oj Mathematics

( a l t h o u g h I d i s l i k e t h e s e x i s t t i t l e , w h i c h i s a l s o m i s l e a d i n g s i n c e t w o o f t h e m a t h e m a t i c i a n s a r e w o m e n , - t h e b o o k w a s w r i t t e n i n 1937).

A p p a r e n t l y o t h e r h i s t o r i a n s o f m a t h e m a t i c s h a v e d r a w n t h e i r i n f o r m a t i o n f r o m B e l l , s o I w i l l n o t m e n t i o n t h e m b y n a m e h e r e . A l l m y i m p o r t a n t s o u r c e s a r e r e f e r e n c e d i n t h e e n d n o t e s . A d d i t i o n a l l y , I f o u n d t h e a r t i c l e s o f J a c q u e l y n S a v a n i o f P r i n c e t o n U n i v e r s i t y (Princeton

Weekly Bulletin, S e p t e m b e r 6, 1993)

u s e f u l , a n d I t h a n k h e r f o r s e n d i n g m e a c o p y o f a p r o g r a m a i r e d o n t h e B B C a b o u t F e r m a t ' s L a s t T h e o r e m .

I a m i n d e b t e d t o C . J . M o z z o c h i f o r a n u m b e r o f p h o t o g r a p h s o f m a t h e m a t i c i a n s i n v o l v e d i n t h e p r o o f o f F e r m a t ' s L a s t T h e o r e m . V e r y w a r m t h a n k s t o P r o f e s s o r K e n n e t h A . R i b e t o f t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y f o r i n f o r m a t i v e i n t e r v i e w s a n d m u c h i m p o r t a n t i n f o r m a t i o n a b o u t h i s w o r k l e a d i n g t o t h e p r o o f o f F e r m a t ' s t h e o r e m . M y d e e p g r a t i t u d e t o P r o f e s s o r G o r o S h i m u r a o f P r i n c e t o n U n i v e r s i t y f o r h i s g e n e r o s i t y w i t h h i s t i m e i n g i v i n g m e a c c e s s t o s o m u c h

i m p o r t a n t i n f o r m a t i o n a b o u t h i s w o r k , a n d h i s c o n j e c t u r e w i t h o u t w h i c h t h e r e w o u l d b e n o p r o o f o f F e r m a t ' s t h e o r e m . I a m g r a t e f u l t o P r o f e s s o r G e r d F a l t i n g s o f t h e M a x P l a n c k I n s t i t u t e i n B o n n a n d t o P r o f e s s o r G e r h a r d F r e y o f t h e U n i v e r s i t y o f E s s e n , G e r m a n y , f o r p r o v o c a t i v e i n t e r v i e w s a n d i n s i g h t f u l o p i n i o n s . T h a n k s a l s o t o P r o f e s s o r B a r r y M a z u r o f

H a r v a r d U n i v e r s i t y f o r e x p l a i n i n g t o m e i m p o r t a n t c o n c e p t s i n t h e g e o m e t r y o f n u m b e r t h e o r y . A n y e r r o r s t h a t r e m a i n a r e c e r t a i n l y m i n e .

I t h a n k m y p u b l i s h e r , J o h n O a k e s , f o r h i s e n c o u r a g e m e n t a n d s u p p o r t . T h a n k s a l s o t o J i l l E l l y n R i l e y a n d K a t h r y n B e l d e n o f F o u r W a l l s E i g h t W i n d o w s . A n d f i n a l l y , m y d e e p g r a t i t u d e t o m y w i f e , D e b r a .

Index

Abel, Niels Henrik, 59 , 76 -77 abelian

groups, 77 abelian varieties, 77

abstract algebra, 71 algebra, 34-35,

40-42 abstract, 71

Diophantine equations, 33 , 92 , 110

algebraic geometry, 35, 110 algebraic

groups, 83 algebraic numbers, 78

algorists, 45 algorithms, 34

American Mathematical Society,

91-92

analysis, 60

Analysis Situs (Poincare), 81 analytic

functions, 55-56 modular forms, 56 ,

83 -84 ,

93 -94 , 99 -100 , 101 -6 , 110

Annals of Mathematics, 134 Archimedes,

5 , 29 , 30 -32 archimedes' screw, 31

architecture, Golden Section in,

39-40 Aristotle, 26 , 27

arithmetic geometry, 110 Arithmetica

(Diophantus), 6 , 9 , 33 , 43 Ars Magna

(Cardano), 42 astronomy, 58 -59

automorphic forms, 83 , 99 -100

axioms, 29

Babylonian Era, 12-15 , 18 , 21 Bachet,

Claude, 42 -43 Bell, E, T„ 5

Bernoulli, Daniel, 45, 46 Bernoulli,

Johannes, 45 Bernoulli, Nicolaus, 45,

46 Bessel, Friedrich Wilhelm, 56

Bolyai, Janos, 70 Bourbaki, Nicolas, 88 -

91 , 98 Bridges, of Konigsberg, problem

of, 49-50

calculus, 5 , 30 differential, 30

integral, 29 , 30 Cantor, Georg, 23

Cardano, Geronimo, 41-42 Catherine

the Great, empress of

Russia, 48 Cauchy, Augustin-Louis, 68 ,

72 , 76 , 77

Chevalier, Auguste, 76 Cicero, 31

Class Number Formula, 121 -24 , 130

Coates, John, 1 ,4 , 117 , 122, 127

comets, 58 -59 complex analysis, 49 ,

54-56 complex number field, 54

complex plane, 55

functions on, 55-56

periodic functions on, 65-66 , 82 -

84

computers

attempts to prove Fermat's Last

Theorem, 10

numerical analysis, 65

conjectures, 43 Conway, John, 1

cosine, 64 Cossists, 41 -42 cubic

equations, 41-42

Dedekind, Richard, 77 -79 Diderot,

Denis, 48 Dieudonne, Jean, 90—91

differential calculus, 30 Diophantine

equations, 33 , 92 , 1 10 Diophantus,

6 , 9 , 32 -33 , 43 Dirichlet, Peter

Gustav Lejeune, 44 ,

59-61 , 67 , 78 DisQuisitiones

Aritbmeticae (Gauss), 53 ,

59 -60 , 76

Eisenstein, Ferdinand Gotthold, 126

Eisenstein Ideal, 110, 126 Elements

(Euclid), 27 elliptic curves, 92 -94 ,

117 connection with modular forms,

93 -94 , 99 -100 , 101 -6 ,

110 , 119 -27

rational, 101-2

semistable, 119 -27

Shimura conjecture, 100-106

switching, 126 -27

zeta functions of, 99-100 Epsilon

Conjecture, 118

See also Shimura-Taniyama conjecture

equations

cubic, 41-42

Diophantine, 33 , 92 , 110 Mordell's

conjecture, 8 5-87 solution surfaces,

86-87 solving, 35 , 40-42 Euclid,

27 , 29

Eudoxus of Cnidus, 5, 29 -30 Euler,

Leonhard, 43 , 44 , 45-50 Euler

System, 122, 130, 132-33

factorization, 66 Faltings, Gerd, 87 ,

107 Fermat, Clement Samuel de, 7

Fermat, Dominique, 5 -7 Fermat,

Pierre de, 4 -8 , 9 , 43 , 44 ,

134-36 Fermat's Last

Theorem

attempts to prove, 9 -10 , 43-45,

66 , 68 -69 corollaries, 43

Gauss on, 53-54 implication of

Mordell's

conjecture, 87 marginal note, 6 ,

9 -10 , 33 , 136 partial proofs,

44 -45 , 51 , 59 , 87 -88

prizes offered for proof, 52 , 69 ,

70 related to Diophantine

equations, 1 10 Shimura-

Taniyama conjecture

and, 4 , 101 , 1 1 1 -14 , 115-

16,

118 -27

Sophie Germain's Theorem, 57 Wiles'

proof, 4 , 10-11 , 118 -27, 132-34

Ferro, Scippione del, 42 Fibonacci

(Leonardo of Pisa), 35-36 Fibonacci

numbers, 25 , 36-37 See also Golden

Section

Fibonacci Society, 39 , 40 fields,

121

first law of hydrostatics, 30 -31

Flach, Matthias, 122, 124 Four

Color Map problem, 50 Fourier,

Joseph, 61 -65 Fourier analysis, 65

Fourier Series, 62 , 64 -65 French

Academy of Sciences, 69 Frey,

Gerhard, 109-14, 115, 119 Frey

curve, 112, 115 functions

analytic, 55-56

automorphic forms, 83 , 99-100

on complex plane, 55-56

Fourier Series, 62 , 64 -65

modular forms, 56 , 83-84 ,

93 -94 , 99 -100 , 101 -6 ,

110 periodic, 62-66 , 82 -84

trigonometric, 64 zeta, 98 , 99 -

100

Galois, Evariste, 71 -76 , 121 Galois

Theory, 72 , 74 , 79 , 1 10 , 120 -21

Gauss, Carl Friedrich, 51 -54 ,

55-59, 61 , 77 , 78 , 92 ,

126 genus, 85-87 geometry

algebraic, 35 , 110

arithmetic, 110

Euclidean, 29

method of exhaustion, 29 -30

non-Euclidean, 70 -71 , 83

origins, 27-28

Pythagorean Theorem, 18 -19 ,

21 , 36 theorems, 28 -29

Germain, Sophie, 56 -58 God, proof

of existence, 48 Golden Section, 25-

26 , 36-40 Guthrie, Francis, 50

Heiberg, J. L., 32 Herodotus, 27-28

Hieron II, king of Syracuse, 30 -31

Histories (Herodotus), 27 -28

Horizontal Iwasawa Theory, 121,

132 -33

hydrostatics, first law of, 30-31

ideal numbers, 68 , 78 ideals, 70 ,

78 -79 , 115, 126 imaginary

numbers, 48 , 49 , 54-56 infinite

descent, 44 integral calculus, 29 ,

30 Ireland, Kenneth F, 114

irrational numbers, 21-23 Iwasawa

Theory, 1 ,117 Horizontal, 121,

132 -33

Al Jabr Wa'l Mucfabalab (Al

Khowarizmi), 35

Katz, Nick, 123-24, 129-30 Al

Khowarizmi, Mohammed Ibn

Musa, 34 -35 Konigsberg,

Seven Bridges of,

49 -50 Kronecker, Leopold,

23 Kummer, Ernst Eduard, 66 -70

Lame, Gabriel, 44 , 66 Lang, Serge,

103, 107-9 Lebesgue, Henri, 44

Legendre, Adrien-Marie, 44 , 59

Leibniz, Gottfried Wilhelm von, 5

lemas, 43

Leonardo of Pisa (Fibonacci), 35 -36

Liber Abaci (Fibonacci), 36 Liber

Quadratorum (Fibonacci), 35 Liouville,

Joseph, 66 , 76 Lobachevsky, Nicolas

Ivanovitch, 70

map-coloring problems, 50 Marcellus,

31

Mazur, Barry, 8 , 79 , 110, 113,

114 -16 , 126 Mesopotamia,

12-15 Mestre, J.-F., 111 method of

exhaustion, 29-30 The Method

(Archimedes), 32 modular forms, 56 ,

83 -84 connection to elliptic curves,

93-94 , 99-100, 101-6 ,

110,

119-27

Shimura conjecture, 100-106

modularity, 94 Mohammed, 34

Monge, Gaspard, 61 , 62 Mordell,

Louis J., 84 -87 Mordell's conjecture,

8 5-87 , 107 music, harmony in, 26-

27

Newton, Sir Isaac, 5

number systems

Babylonian, 13 -15 base 10 , 26

number theory, 52 -53 , 58 , 1 10 ,

114 algebraic numbers, 78

Fermat's theorems in, 9 Fourier

series used in, 65 prime numbers,

8 , 44 , 60 , 68 -70

numerical analysis, 65

Olbers, H. W. M., 53-54 Ovid, 27

Pacioli, Luca, 41 Parthenon, 39

pentagons, 24 -25 , 26 perfect

numbers, 17 periodic functions, 62-

65 extended to complex plane, 65-

66 , 82-84 Philolaos of Tarentum, 24

pi, 12 , 21 Plato, 24 , 29

Poincare, Henri, 79 -84 , 101 Poisson,

Simeon-Denis, 74 prime numbers, 8 ,

60

as exponents in Fermat's Theorem,

44

irregulars, 68 -70

proofs

of Fermat's Last Theorem, 4 ,

10 -1 1 , 118 -27 , 132 -34 of

Frey's conjecture, 115-16 of

God's existence, 48 of Shimura-

Taniyama

conjecture, 118 -27 of theorems, 28 -

29 pyramids, 16 , 21 , 40

Pythagorean Pentagon, 24 -25 , 26

Pythagorean Theorem, 18 -19 , 21 ,

36 Pythagorean triples, 13 -16 , 19 ,

35 Pythagoreans, 17 , 18 -22 , 24-

25, 26 -27

Pythagorus, 13 , 15-18 , 22 , 24

rational numbers, 22 -23 Ribet,

Kenneth, 3 -4 , 110 , 111 -12, 113 ,

1 14 -16 , 129, 136

Sarnak, Peter, 10-1 1 , 125

semistable elliptic curves, 119 -27

Serre, Jean-Pierre, 97 , 98 , 102,

105,

107, 108 , 1 1 1 , 1 1 8

Seven Bridges of Konigsberg,

problem of, 49-50 Shimura,

Goro, 65 , 96-97 ,

100-101, 107-8 Shimura-

Taniyama conjecture, 65 , 100-106

proof of, 118 -27 used to prove

Fermat's Last

Theorem, 4 , 101, 1 1 1 -14 , 115-

16, 118 -27 Weil's association with,

102-9 sine, 64 , 82 square roots

computing, 65 of five, 26

of negative one, 54-55

of two, 21-22 squared numbers,

13-16 , 19 , 20 , 35 statistics, 53

Stewart, Ian, 37

Taniyama, Yutaka, 95-100

Taniyama-Shimura conjecture. See

Shimura-Taniyama

conjecture Tartaglia, Niccolo,

41-42 Taylor, Richard, 130-31,

133 -34 tetraktys, 26-27

theorems, 28 -29 Tokyo-Nikko

Symposium on

Algebraic Number Theory, 96-100

topology, 49 , 81-82

map-coloring problems, 50 Mordell's

conjecture, 8 5-87 torus, 85

triangles, Pythagorean Theorem,

18 -19 , 2 1 , 36

trigonometric functions, 64

Weil, Andre, 90 , 91 , 97-98 , 99 -

100 , 102 -9 , 120 Weil Curves, 106

Weil-Taniyama conjecture, 104, 106

See also Shimura-Taniyama

conjecture Wiles, Andrew, 110,

1 1 3 interest in Fermat's Last

Theorem, 116 -18 lectures at

Cambridge

conference, 1 -4 , 127 peer

review of proof, 10 , 127 -30

problem found in proof, 10-1 1 ,

129 -32

proof developed by, 118 -27 , 132 -

34 Wolfskehl Prize, 70

zeta functions, 98 , 99 -100