Microchip Mathematics Number Theory

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Computational Number Theory

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  • Microchip Mathematics number theory for computer users

    Keith Devlin lIathematlcs Department Unlvenlty of Lancaster

  • SHIV A PUBLISHING LIMITED 64 Welsh Row, Nantwich, Cheshire CW5 5ES, England

    Keith Devlin, 1984

    ISBN 1 850140472

    All rights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted in any form or by any means. electronic, mechanical, photocopying. recording and/or otherwise, without the prior written permission of the Publishers.

    This book is sold subject to the Standard Conditions of Sale of Net Books and may not be resold in the UK below the net price given by the Publishers in their current price list.

    The front cover shows the author with a print-out of the largest known prime number, a number with 39751 digits. The print-out is 9 feet in length. It required over half an hour of main frame computer time to work out the digits in this number. (Photograph taken at The Computer Unit, Warwick University, courtesy of Dr Keith Halstead.)

    Printed and bound in Great Britain by Billing and Sons Limited

  • Contenls

    O.

    I.

    PREFACE

    BACKGROUND: PRIME NUMBERS

    1.

    2.

    3.

    4.

    5.

    Prime Numbers

    The Sieve of Eratosthenes

    The Distribution of Primes

    Largest Known Primes

    Conjectures About Primes

    Exercises 0

    Computer Problems 0

    BASIC CONCEPTS

    1.

    2.

    3.

    4.

    5.

    Mathematical Induction

    Divisibility. The Euclidean Algorithm

    Efficiency of Algorithms. Mu1tiprecision

    Arithmetic

    The Fibonnaci Sequence and the Efficiency

    of the Euclidean Algorithm

    Prime Numbers

    v

    1

    2

    4

    5

    8

    9

    11

    12

    14

    14

    23

    34

    43

    48

    iii

  • II.

    III.

    iv

    6. Diophantine Equations

    Exercises I

    Computer Problems I

    CONGRUENCES

    1.

    2.

    3.

    4.

    Congruence

    Modular Arithmetic

    Fermat's Little Theorem and the

    Euler Phi-Function

    Random Number Generators and Primitive Roots

    Exercises II

    Computer Problems II

    PRIMALITY TESTING AND FACTORISATION

    1.

    2.

    3.

    4.

    Perfect Numbers and Mersenne Primes

    Public Key Cryptography

    Primality Testing

    Factorisation Techniques

    Exercises III

    Computer Problems III

    RECOMMENDED FURTHER READING

    INDEX OF NUTl\1'fON

    INDEX

    51

    55

    59

    62

    62

    77

    94

    107

    128

    135

    138

    139

    153

    163

    178

    191

    197

    202

    203

    204

  • Preface

    In the Autumn of 1983, in the face of the phenomenal growth of

    home computer sales in the U.K., the national British newspaper

    The Guardian decided to produce, each week, a 'Computer Page'.

    Noone was quite sure exactly what should go into the page on a

    regular basis, but it was thought that a fortnightly column on

    computer mathematics might be a good idea, and when the computer

    page first appeared on 20th October of that year, it included a

    small item on binary arithmetic by me.

    From the mail I received after my column had been running

    for a few months, it was clear that the microcomputer age had brought

    with it a huge increase in the number of (potential)

    'recreational mathematicians'. Though in many cases without any

    formal training in mathematics, my correspondents displayed tremendous

    mathematical ability, and I was frequently asked if I could recommend

    any suitable books. What they seemed to want was a genuine

    mathematics text book, but one which did not require a great deal

    of prior knowledge. This is intended to be just such a book.

    Number Theory is one of the few areas of modern mathematics

    which is accessible to the non-expert. (At least, the kind of

    Number Theory considered here: there is a lot of other material

    v

  • which also goes under the title 'Number Theory', most of which

    is pretty well inaccessible to the majority of trained

    mathematicians~) It is also an area in which there is a genuine

    two-way flow between man and the computer. Indeed, it was this

    fascinating interplay of brain power and computer power that

    awakened my own interest in the subject to a level where I began

    to give a course on the subject at Lancaster University and,

    coincidentally, write about it in The Guardian. (Previously my

    mathematical research work had been in Set Theory, a subject dealing

    almost exclusively with the mysterious world of the infinite.)

    This is a book about (the computational aspects of) Number

    Theory. Though written for university undergraduates in

    mathematics, I have tried to present the material in such a way

    that it can be followed by the keen but largely untrained 'amateur'

    sitting at home with (or possibly even without) a cheap home

    computer. I do not pretend to give a complete coverage of the

    computational aspects of Number Theory. (For instance, no mention

    is made of Quadratic Reciprocity, a tremendously important part

    of the subject.) Rather my aim is to cover the (very) basic parts

    of Number Theory and at the same time give some indication of the

    way in which Number Theory both feeds off and leads to advances

    in Computation Theory. Consequently, if the book were used as

    a text to accompany a university lecture course, the lecturer would

    presumably deal with additional topics not covered in this book.

    In writing this book, I made extensive reference to, in

    particular, two excellent books, to which this text could be regarded

    as a precursor. David Burton's book Elementary Number Theory

    gives a wonderfully readable coverage of (essentially the non-

    vi

  • computational aspects of) Number Theory, and covers many more topics

    than I have space for here, whilst Donald Knuth's 'The Art of

    Computer Programming, Volume ~' is the 'bible' for serious

    computational number theorists.

    The book is structured in a way that assumes a more or less

    direct passage from start to finish, though an index is provided

    to enable the book to be used as a reference text if necessary.

    Each chapter (including an informal preparatory chapter) ends with

    a selection of (mathematical) Exercises, grouped according to the

    section they refer to, and some Computer Problems. The latter

    are, for the most part, just initial 'pointers' as to what can be

    tried out on a computer, and I would hope that these are enough

    to spur the reader on to carrying out further computer investigations

    of his or her own devising.

    To assist readers who wish to skip proofs and concentrate on

    the development of the main results, the symbol 0 is used to

    indicate the end of a proof. (Whenever this symbol occurs

    immediately following the statement of a result, this indicates

    that the proof is so obvious as to require no further comment.)

    For easy reference, all results obtained are numbered consecutively,

    the reference numbers consisting of the Chapter number, section

    number, and result number.

    Keith Devlin

    Lancaster, August 1984

    vii

  • Pierre De Fermat: 'The Father of Number Theory'. Born in 1601

    near Toulouse in France, Fermat was a jurist by profession, and

    only took up mathematics as a hobby in his thirties. Through

    correspondence with many of the leading scholars of the day, Fermat

    developed most of the pivotal ideas of present day Number Theory.

    Many of his ideas to simplify mental calculation are nowadays

    employed to speed up computer algorithms. This painting is from

    the collection of the Academie des Sciences, Inscriptions et

    Belles Lettres de Toulouse; it is reproduced here with the kind

    permission of Robert Gillis.

  • o Background: Prime Numbers

    Numbers constitute the one mathematical system familiar to all

    mankind, at least if by 'number' you mean 'positive whole number'

    as did the Ancient Greeks. Today the professional mathematician

    uses the phrase 'natural number' to denote the positive whole

    numbers 1,2,3, .. This is a book about these 'natural' numbers,

    and we shall rarely have occasion to speak of other numbers such

    as proper fractions like t,t, or t. The study of the natural

    numbers is known as 'Number Theory', and in keeping with the

    traditions of that subject we shall use the word 'number' to mean

    'natural number' unless otherwise indicated. (This convention

    is used in the very name 'Number Theory' of course.)

    The natural numbers are so fundamental to the rest of

    mathematics that the famous 19th Century mathematician Leopold

    Kronecker once remarked that 'God created the natural numbers,

    and all the rest is the work of man.' What he meant by this

    was that, starting from the natural numbers it is possible to

    construct, in a rigorous fashion, the entire edifice of modern

    mathematics, which is true, and that the natural numbers themselves

    cannot be constructed (in a mathematical sense) from any simpler

    entities, which was true when Kronecker made his remark but is

    1

  • no longer valid, Cantor's Set Theory having p