Fermat's Last Theoremand Modern Arithmetic

Pierre de Fermat's famous conjecture may have been proved at last. Ironically,it turns out to be a marginal note in a body of work with wider significance

Kenneth A. Ribet and Brian Hayes

Eric Temple Bell, the mathematicianand biographer of mathematicians,believed that Fermat's last theoremwould be one of the questions left unresolved when human civilization destroyed itself in nuclear war. Bell madethis prediction shortly before his owndeath in 1960. If he had lived a fewdecades longer, it is an interestingquestion whether he would have beenmore surprised at humanity's continuing survival or at the announcement,on June 23,1993, of a proof of Fermat'slast theorem.

The theorem itself is easily stated.Pierre de Fermat asserted that if a, band c are integers greater than 0, and ifn is an integer greater than 2, thenthere are no solutions to the equation:

a" + b" = c".

The simplicity of the statement is deceptive: The proposition resisted all attempts at proof for more than 350 years.And the recent proof, devised by Andrew Wiles of Princeton University, requires an extraordinary arsenal of mathematical tools and techniques to attack

Kenneth A. Ribet is professor of mathematics atthe University of California at Berkeley. Hereceived his A.B. and A.M. degrees from BrownUniversity in 1969 and his Ph.D. from HarvardUniversity in 1973. Ribet has -worked on diversequestions in number theory and arithmeticalgebraic geometry; his best-known result is hisproof that the Taniyama-Shimura conjectureimplies Fermat's last theorem. In 1989 he receivedthe first Prix Fermat (jointly with Abbas Bahri).Brian Hayes is a science writer and former editorof American Scientist; his contributions to thisarticle concern the exposition, not the mathematics.Address for Ribet: Department of Mathematics,University of California, Berkeley, CA 94720.Internet: ribet@math.berkeley.edu.

the problem. Wiles's proof is embodiedin a dense and difficult manuscript,which incorporates by reference a vastlylarger body of mathematical work developed over the past 30 years or more.

As this article is being written, thestatus of Wiles's proof is uncertain.Checking by referees revealed a fewproblems, most of which were quicklyresolved, but one gap in the argumentappears to be more serious. Wiles hassaid he is confident the gap can befilled, but until the proof is publishedand reviewed by the larger mathematical community, the issue will remainunsettled. Indeed, the proof is not aproof and the theorem is not a theoremuntil all the outstanding problems havebeen resolved. (Nevertheless, in this article we shall continue to speak of theproof and the theorem as a matter ofconvenience.)

It is important to understand thetrue place of Fermat's last theorem inmodern mathematics: It is a much-celebrated puzzle, but it is hardly a central or crucial proposition. Having aproof of the theorem may not lead tomuch else of great interest. On the other hand, the pursuit of a proof has contributed to the development of muchimportant mathematics. In particular,Wiles approached the problem by setting out to prove another proposition,called the Taniyama-Shimura conjecture, from which Fermat's last theoremfollows as a corollary.

The Taniyama-Shimura conjecture isdeeper and potentially more significantthan Fermat's last theorem itself. It belongs to a realm of mathematics thathas been developing rapidly over thepast three decades without attractingmuch notice outside the mathematics

profession. This realm is called "arithmetic algebraic geometry," or "modernarithmetic." It grew out of an attempt toapply the methods of modern mathematics to the study of problems, calledDiophantine problems, in which thegoal is to find all solutions in wholenumbers to a family of equations. Modern arithmetic has a rich structure of itsown, and it seems to be connected inone way or another to every otherbranch of mathematics. It is remarkablethat the abstract machinery of this discipline has led to a new understandingof the most famous of all Diophantineproblems—Fermat's last theorem.

MarginaliaThe story of how Fermat proposed his"last theorem" has been told manytimes, but it is too good a story to forgotelling it again. Pierre de Fermat wasbom in the south of France in 1601 andspent most of his life in Toulouse, wherehe was a prominent jurist in the bureaucracy serving Louis XIV. As a mathematician he was an amateur, but a well-connected one; he carried on extensivecorrespondence with Rene Descartes,Blaise Pascal and other luminaries of theage. Indeed, the main source of knowledge about his mathematical work is hiscorrespondence—and his annotationsin the margins of books.

Sometime in the 1630s Fermat wasreading the Arithmetic of Diophantusof Alexandria, a work probably written in the third century C.E., whichdiscusses various problems to besolved in whole numbers or in rationalnumbers (ratios of whole numbers).Fermat made numerous notes in hiscopy of the Arithmetic; the particularmarginal comment of interest here per-

144 American Scientist, Volume 82

20,000

Figure 1. Fermat's last theorem, which asserts that the equation a" + b" = c" has no integer solutions when » > 2 and abc * 0, can be given asimple geometric interpretation. For any given value of n, the function f(a,b) = a" + b" defines a smooth surface in three-dimensional space.In the case of » = 1 (upper left) the surface is a plane that passes through infinitely many points with integer coordinates; indeed, at everypoint where a and b are integers, the third coordinate a + b is also an integer. Integer points where a or b is equal to zero are marked by blackdots; other integer points are shown as red dots. For » = 2 (upper right) the surface is a paraboloid, and the only integer points are those thatsatisfy the Pythagorean equation a2 + b2 = c2. Apart from points along the axes where a = 0 or b = 0, there are only four such points on thesmall section of the surface visible here; they correspond to the equations 32 + 42 = 52 and 62 + 82 = 102. On the entire surface, the number ofinteger points is infinite. Sections of the surfaces for tt = 3 (lower left) and ;/ = 5 (lower right) have no integer points (except along the a = 0and b = 0 axes). Fermat's last theorem implies these surfaces can be extended to infinity without ever intersecting a point with three integercoordinates. Moreover, the theorem states that the same is true of the surfaces corresponding to all other values of « greater than 2.

tained to Question 8 in Book 2, whereDiophantus asked, "Given a numberthat is a square, write it as a sum of twoother squares." Fermat's note, translated from the Latin, reads: "It is impossible to separate a cube into two cubes or

a fourth power into two fourth powersor, in general, any power greater thanthe second into powers of like degree. Ihave discovered a truly marvelousdemonstration, which this margin istoo narrow to contain."

The tantalizing suggestion of a proofonce known and then lost to posterityhas doubtless contributed to the popular romance of Fermat's last theorem.So has the designation "last," althoughFermat had nothing to do with that.

1994 March -Apr i l 145

The theorem was surely not the lastone he proposed in his lifetime; helived on until 1665 and made manyfurther contributions to mathematics.The label "last" arose in the 18th or19th century and was apparentlymeant to identify the theorem as thelast of Fermat's propositions to remainneither proved nor disproved.

Did Fermat really have a "marvelous" proof that he could have written out if only the margin had been alittle wider? The question is another onethat has a good chance of outlasting human civilization. A likely answer is thatFermat thought he had a proof but laterdiscovered a flaw in it. In subsequentletters to colleagues he referred toproofs of the specific cases where ;/ = 3and ;/ = 4, but the general proof wasnever mentioned again.

Early EffortsThere is no trouble finding integer solutions to a" + b" = c" when n is 1, sincethe equation then reduces to the simpleform a + b = c. Because the sum of anytwo integers is also an integer, for any aand b there is always a c satisfying the

equation. When n is equal to 2 (the caseconsidered by Diophantus) the problem is only a little harder. The equationa2 + b2 = c2 is of course the Pythagoreanformula for the sides and hypotenuseof a right triangle; it has infinitelymany integer solutions, starting withthe familiar 32 + 42 = 52. Euclid, centuries before Diophantus, gave amethod for generating all sets of suchPythagorean triples.

Given the infinity of solutions whenn = 1 or n = 2, it seems surprising thatthere should be no integer solutionswhatever for all n > 3, but that is Fermat's assertion. The theorem has a geometric interpretation. For each value ofn, the equation a" + b" = c" defines asurface in three-dimensional space.The surfaces for n = 1 and ;/ = 2 passthrough infinitely many points thathave three integer coordinates, but thesurfaces for all higher values of n passthrough no such points (except alongthe planes where a = 0 and b = 0).

Fermat himself proved the theoremfor the case;/ = 4 (and this time he wrotedown his argument, in another marginalnote). In fact Fermat proved a slightly

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Figure 2. Famous marginal note in which Pierre de Fermat stated his "last theorem" is preserved in an edition of the Arithmetic of Diophantus published by Fermat's son Samuel.Fermat read the Arithmetic in the first modern edition, published in 1621 by Claude-GasparBachet, but the copy with his annotations has not been found. In Samuel Fermat's editionthe note is transcribed below the Latin and Greek texts of Question 8 in Book 2. It reads: "Itis impossible to separate a cube into two cubes or a fourth power into two fourth powers or,in general, any power greater than the second into powers of like degree. I have discovered atruly marvelous demonstration, which this margin is too narrow to contain." (Photographcourtesy of the Duke University Special Collections Library.)

more general proposition, showing thatthere are no integer solutions to the equation rt4 + fr4 = c2; since any perfect fourthpower is also a perfect square, this resultimplies the truth of the original theoremfor n = 4. Put another way, Fermatshowed that there are no Pythagoreantriples a2 + b2 = c2 where a and b arethemselves perfect squares. The idea underlying the proof is a technique Fermatinvented, called the method of infinitedescent. Begin by assuming there are indeed integer solutions of the equationrr* + fr4 = c2_ Fermat found a sequence ofoperations that, given any such solution, generates a smaller one. From thenew solution the same sequence of operations yields a still smaller solution.The process can be continued withoutlimit, creating an infinite series of ever-smaller solutions. But such a series ofcontinually diminishing numbers cannot exist in the positive integers, whichhave a well-defined lower bound (thenumber 1). The only way to avoid theimpossibility is to abandon the initial assumption that an integer solution exists.

The case of n = 3 was undertaken byLeonhard Euler, the great 18th-centurySwiss mathematician. His proof also relies on infinite descent, but it is moreconvoluted than the proof for n = 4. Inthe years that followed, several moreindividual cases of Fermat's last theorem were proved. In the 1820s theFrench mathematician Adrien-MarieLegendre and the German P. G. LejeuneDirichlet produced proofs for n = 5.Dirichlet went on to attempt a proof forn = 7, but he was able to complete theproof only for n = 14; a proof for n = 7was subsequently devised by GabrielLame of France. Then in 1847 a majoradvance was reported by the GermanErnst E. Kummer, who came tantaliz-ingly close to a general proof. Rummer's work implies that Fermat's lasttheorem is true for an infinite class ofexponents, namely all values of 11 thatare divisible by "regular" primes,which are a subset of the prime numbers (positive integers divisible only bythemselves and by 1). The only non-regular primes less than 100 are 37, 59and 67, and Kummer was subsequently able to supply a proof for these specific values as well. Thus Fermat's lasttheorem was proved for all n < 100.

In recent years computer-aidedstudies have pushed the lower limitof any possible counterexample successively higher. A result publishedlast July (Buhler, Crandall, Ernvall

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and Metsankyla 1993) implies that Fermat's last theorem must be true for allexponents n less than 4 million, and soany integer solution to a" + b" = c"would have to consist of astronomically large integers. (It turns out the smallest possible value of c" would be a number of more than 26 million decimaldigits.) Still, no mathematician couldconsider the question settled just becausea finite range of cases has been dismissed. There are infinitely many ofKummer's nonregular primes, and so noextension of the case-by-case analysis canever be complete.

What Needs to Be ProvedThe modern approach to Fermat's lasttheorem is an indirect one. It does not attack Fermat's equation a" + b" = c" itselfbut instead analyzes a new equation of adifferent form, in which the numbers a"and b" have an essential role.

In broad outline, the argument goesas follows. Suppose there is a counterexample to Fermat's last theorem, or inother words a pair of integers a" and b"whose sum is also a perfect nth power.Then there must exist a mathematicalobject called an elliptic curve specifiedby an equation whose coefficients aredetermined by a" and b". Call the elliptic curve E. One of us (Ribet) proved in1986 that the curve E cannot have a certain property called modularity. WhatWiles announced last June is that all elliptic curves in a class that includes £are modular. From this contradiction itfollows that E cannot exist, and neithercan the supposed counterexample toFermat's last theorem.

In this article we shall fill in a fewdetails of this argument. In particularwe shall explain what an elliptic curveis and what it means to be modular. Arigorous account of the entire proofand its background would be very demanding indeed, and so we shall onlyilluminate the main points.

It is best to begin by refining somewhat the question of what is to beproved. Specifically, several constraintscan be put on the values of a, b, c and nin the Fermat equation a" + b" = c".First, we can confine our attention tocases where n is an odd prime number.Only prime exponents need be considered because any counterexample tothe theorem in which n is compositewould imply the existence of a smallercounterexample with a prime exponent. In other words, if aP'l + bP'i = c'"'has a solution in integers, then so must

Figure 3. Pierre de Fermat (1601-65) was a successful lawyer and judge, but he is knowntoday almost entirely for his activities in a realm where he was an amateur: mathematics.Although he did not publish his mathematical work, he carried on an extensive correspondence with other savants, and made major contributions to number theory and the studyof probability. This engraving appears in his son's edition of Diophantus.

both aP + bP = cP and at + bi = cl Theonly composite values of n that escapethis reasoning are powers of 2, sincethey have no odd factors. However, allpowers of 2 greater than the first power are divisible by 4, and Fermat's ownproof dispenses with this case. Indeed,strictly speaking there is no need toworry about any multiples of 3, 4, 5, 7and all primes less than 4 million, butexploiting these known results offersno advantage in devising a generalproof. Wiles's proof applies to all primevalues of n greater than or equal to 5.

A similar line of argument eliminates all cases except those where a, band c are relatively prime—that is tosay, they have no factors in common.Again, if you knew a counterexamplein which a, b and c had a common fac

tor, you could divide both sides of theequation by that factor to produce asmaller solution.

We shall state two further factsabout the values of a", b" and c" without explaining them in detail. Exactlyone of a, b and c must be even; we shallassume that the even number is b.Since /? is at least 5, b" must be divisiblenot only by 2 but also by 25, or 32.Among the two odd values, one mustbe congruent to 1 modulo 4 (that is, itmust leave a remainder of 1 when divided by 4), and the other must be congruent to 3 modulo 4. Here we shall assume that a" is 3 modulo 4.

From here on we can essentiallyleave behind Fermat's equation andwork with new variables A, B and C,which represent n", b" and c" respec-

1994 March-April 147

tively. The new variables must satisfyall the constraints we have just established. In particular:

A + B = CABC*0A, B and C are relatively primeB is divisible by 32A = 3 modulo 4 and C = 1 modulo 4.

A final property of A, B and C should notbe left unstated: To serve as a counterexample to Fermat's last theorem, A, Band C must be perfect nth powers,where n is a prime >5. Hence the product ABC is also a perfect nth power,since n"b"c" = (abc)".

It is at this point that elliptic curvesenter the story. The curve of interest isdescribed by the equation

y2 = x(x-A)(x + B)where the numbers A and B are derived from the hypothetical counterexample to Fermat's last theorem introduced above. (Although C does notappear in the equation, no informationis lost because C can be represented by

y2 = x(x-3)(x+32)y2 = *3 + 29x2-96x

Figure 4. Elliptic curve is the locus of pointsthat satisfy a certain cubic equation. Suchcurves have a deep connection with Fermat'slast theorem. Specifically, if there were acounterexample to the theorem, it wouldimply the existence of an elliptic curve withsome highly distinctive properties. Thecurve shown here is defined by the equationy2 = x3 + 29.v2 - 96.v, or in factored formy2 = x(x-3)(x + 32).

the sum A + B.) The strategy of theproof will be to show that the curvespecified by this equation has the property that it is not modular, and then toshow that all elliptic curves in a classthat includes this one must be modular.The only escape from the contradictionis to acknowledge that numbers with allof the properties attributed to A, B andA + B cannot exist.

Elliptic CurvesBefore proceeding with our account ofthe proof, we must pause to introducethese interesting mathematical objectscalled elliptic curves. To begin with, anelliptic curve is not an ellipse. Thename reflects a connection with ellipticfunctions (which were devised to helpcalculate the perimeter of an ellipse,but which have turned out to have other uses as well). Elliptic curves areplane curves defined by a certain classof cubic equations, and their shape isnot even vaguely elliptical.

We can construct a specific ellipticcurve by choosing values for the numbers A and B in the equation givenabove. The constraints on these numbers require that A be congruent to 3modulo 4, and so a straightforward assignment is A = 3. Similarly, B must bedivisible by 32, and so a logical choiceis B = 32. (Of course A = 3 and B = 32do not actually constitute a counterexample to Fermat's last theorem; theyare merely numbers that satisfy someof the criteria that would have to bemet by any such counterexample.)With these substitutions, the equationbecomes:

y2 = x(x-3)(x + 32).Multiplying the three factors on theright-hand side of the equation yieldsan equivalent expression:

i/2 = .\-3 +29x2-96x.Note that this is a cubic equation, or

an equation of degree 3, since the highest power of any variable is a cube;more specifically the equation relatesthe square of y to a cubic polynomial inx. Also note that the coefficients of theequation are integers; in general the coefficients of an elliptic curve can benumbers of any kind, but the curveswe shall consider here will be assumedto have integer coefficients unless stated otherwise.

The equation we have just constructed defines a curve in the x, y plane. Thecurve is the locus of all points whose x

and y coordinates satisfy the equation.For example, the point (0, 0) is clearlyon the curve, since the substitutionx = 0, y = 0 makes the equation a truestatement. The curve defined by theequation is sketched in Figure 4. Itconsists of two disconnected pieces: aclosed loop to the left of the y axis andan infinitely long open segment to theright. The precise form of the curve depends on the values of the coefficients,and in some cases it is a single connected piece.

Not every cubic equation generatesan elliptic curve. To qualify, a curvemust be smooth, or nonsingular, a concept that can be made more precise bysaying that the curve must have a well-defined tangent at every point along itslength. The curve may not have anycusps, where there is no defined tangent; nor may it cross itself, creating anodal point with two or more tangents.Figure 5 shows a few examples ofsmooth elliptic curves and of other cubic curves having singularities.

From an algebraic point of view, requiring that the curve be nonsingularis equivalent to requiring that theequation have three distinct roots, or inother words that three different valuesof x make the expression x(x - A)(x + B)equal to zero. Obviously one of theroots is x = 0, and the others are x = Aand x - -B. Hence to generate a properelliptic curve, the values of A and B mustsatisfy three constraints: A * 0, B * 0 andA * -B. The last constraint can be rephrased as A + B * 0, or equivalentlyC * 0, and so the overall requirementis ABC 5*0.

Why have mathematicians focusedso much attention on this one familyof curves? After all, there are infinitelymany polynomial equations in x and y,generating an endless variety of curvesin the plane. What is so special aboutthe elliptic curves? One answer is thatelliptic curves are the first nontrivialfamily of curves from the Diophantinepoint of view.

All plane curves—or the equationsthat generate them—can be classifiedaccording to their genus, which is anumber closely related to the equation'sdegree. Specifically, a nonsingular curvedefined by an equation of degree d has agenus equal to (d - l)(d - 2)/2. Linesand conic sections—ellipses, parabolasand hyperbolas—are defined by linearequations (d = 1) or quadratic equations(d = 2) and necessarily have genus 0. Elliptic curves, which are nonsingular by

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definition and have degree 3, are curvesof genus 1. Nonsingular equations ofthe fourth, fifth or greater degree yieldcurves of higher genus. In 1922 Louis J.Mordell made an intriguing observation relating the genus of an equationto the number of rational solutions ithas, or equivalently the number ofpoints with rational coordinates thecurve passes through. It was alreadyknown that curves of genus 0 alwayshave either no rational solutions or infinitely many, and the infinite cases arealways easy to describe. Mordell conjectured that all curves of genus 2 orgreater have at most finitely many rational solutions. In 1983 Mordell's conjecture was proved (to the surprise ofthe mathematical community) by GerdFairings, a young mathematician thenat the University of Wuppertal in Germany. Curves of genus 1—namely theelliptic curves—remain as an intermediate case with no simple way of tellingwhether the number of solutions is finite or infinite.

Chords and TangentsOn an elliptic curve, the number of rational points can be either finite or infinite, depending on the coefficients ofthe specific equation. In all cases, however, the set of rational points has a richstructure, allowing them to be exploredsystematically.

If you draw a line, or chord, betweenany two points on an elliptic curve—whether the points are rational or irrational—the line can be extended to intersect the curve at a third point. Someobvious exceptions to this rule arechords drawn between points thathave the same x coordinate, so that thechords are parallel to the y axis. These"vertical" chords extend to infinitywithout striking the curve again. Theexceptional nature of the verticalchords can be eliminated by adding tothe curve a single extra point at infinity,where all the vertical chords can beimagined to converge. This extra pointis called the origin and is designatedby the symbol O; any vertical chordmust pass through O. Thus when theorigin is made a part of the curve,every chord intersects the elliptic curveat exactly three points.

There is a similar construction forlines drawn tangent to an ellipticcurve: Every tangent intersects thecurve at one point in addition to thepoint of tangency. In the special case ofvertical tangents, the additional point

elliptic curves

y2 = x3-x

cubic curves with singularities

y2 = *3

BTjJjc

/

Figure 5. Cubic equations define curves of various forms, only some of which qualify aselliptic curves. The curves shown in both of the upper graphs are elliptic curves; one ofthem consists of two separate pieces, whereas the other is a single connected strand. Thecurves in the lower graphs are not elliptic because they have singularities, or points wherethe curve does not have a unique tangent. Both curves are singular at the point (0, 0).

of intersection is the origin O. A tangent can be thought of as the limitingcase of a chord that gets shorter andshorter until finally its end points coincide. In effect, then, the tangent is achord that passes through the samepoint twice. According to this line ofreasoning, every chord or tangentdrawn on an elliptic curve has exactlythree points of intersection. It is to preserve this property that cubic curveswith singularities are excluded from

the category of elliptic curves, sincecusps or other singularities are pointsthat do not have a unique tangent.

The geometry that makes possiblethe chord-and-tangent construction onelliptic curves becomes more remarkable when attention turns to the rational points on the curve—the pointswhose .V and y coordinates are both rational numbers. When a chord is drawnthrough any two rational points, thethird point of intersection is also ratio-

1994 March-April 149

nal (see Figure 6). Similarly, a tangent tothe curve drawn at a rational pointstrikes the curve at another rationalpoint. (For the purposes of this analysis,the origin O is considered a rationalpoint.) Thus the chord-and-tangentprocedure offers a mechanism for generating rational points: Given any oneor two points known to be rational,there is a direct method for constructing more points. Indeed, it followsfrom a theorem of Mordell that there isa finite set of rational points fromwhich the chord-and-tangent mechanism can generate all the rationalpoints on the curve.

With one further refinement, the setof rational points on an elliptic curvetakes on the structure of a mathematical group. A group consists of a set ofelements along with a "compositionlaw"—a way of combining two elements that always yields another element of the group. The classic exampleof a group is the set of integers withthe operation of addition; adding anytwo integers yields another integer. Agroup must have an identity element,which in the case of integer addition iszero; for any integer n, n + 0 = n. Also,each element n must have an inverse,which combines with n to yield the

identity element; in the case of integeraddition, -n serves as the additive inverse of /;.

The rational points on an ellipticcurve form a group under a composition law just a little more complicatedthan the chord-and-tangent process explained above. The composition procedure works as follows (see Figure 7). Inorder to "add" two points p and q, firstextend the chord between them to findthe third point of intersection, whichcan be labeled r. Now form the chordbetween the origin O and r, and extendthe chord to produce another point, ;■'.This new point r' is the "sum" of pand q. The reason for introducing thepoint O into this operation is that itserves as an identity element. For anypoint p, p + O - p. Thus the groupcomposition law makes it possible toperform a kind of arithmetic on the rational points of an elliptic curve.

An arithmetic operation of particular importance is adding a point to itself. Geometrically, this process is justthe case of point addition that makesuse of a tangent instead of a chord.Arithmetically, it is a group-law analogue of multiplication by an integer.The sum P + P is equivalent to theproduct 2P. Adding P yet again to the

result of these operations yields theproduct 3P, and so on. For some pointsthis process can be continued indefinitely without ever returning to a pointalready visited; such points are said tobe of infinite order. Other points, of finite order, can be added to themselvesonly a finite number of times beforethey yield O as the product; after that,adding P to O yields P again by definition, and the finite sequence of pointsrepeats. Figures 8 and 9 show some examples of finite-order and infinite-order points.

What does the arithmetic of ellipticcurves have to do with Fermat's lasttheorem? The connection is throughthe concept of a curve's being modular,or associated with a modular form,which we turn to next.

What It Means to Be ModularThe study of elliptic curves can betraced back to Fermat and even to Diophantus, and modular forms havetheir origins in the 19th century, but thetwo areas have been deeply linkedonly since 1955. In that year YutakaTaniyama, a young Japanese mathematician, made a bold conjecture,which initially took the form of a seriesof problems presented at a conference.

Figure 6. Chord-and-tangent process illuminates the relations of rational points on an elliptic curve. The procedure works as follows.Given any two points on the curve with rational coordinates, construct a line connecting them and extend it to infinity in both directions.The line will either be a chord that intersects the curve at a third rational point (left diagram), or it will be tangent to the curve at one ofthe selected points (middle diagram). A tangent can be thought of as a kind of degenerate chord that intersects the curve twice at the samepoint, so that the tangent, too, effectively touches the curve at three rational points. The only apparent exceptions to this rule are chordsand tangents parallel to the y axis (right diagram). To cope with these exceptions, an extra point, called the origin, is added to the curve;the origin is a point at infinity, but it is shown here as a line at the top of each diagram. The origin becomes a third point on all verticalchords and tangents, which accordingly have the same geometry as other chords and tangents.

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origin (O)

(-24,72)+ (-12,60) =(8,-40)

Figure 7. Rational points on an elliptic curve form a mathematical group under a slightly more elaborate version of the chord-and-tangentprocess. The group composition law provides a means of "adding" two points (left diagram): First draw a chord connecting them and findthe third point of intersection; then construct another chord between this third point and the origin. The third point on this latter chord isthe sum of the original two points. To add a point to itself (middle diagram), construct a tangent rather than a chord in the first step, thenagain draw a line from the origin through the third point identified by the tangent. The importance of the origin in these procedures isthat it serves as the identity element of the group. Adding any point to the origin (right diagram) yields the same point again.

The conjecture was made more preciseby Goro Shimura of Princeton University, and it is now known as the Taniyama-Shimura conjecture. It contendsthat all elliptic curves with rational coefficients are modular. At the time, thisproposal was viewed with a certainskepticism, but it has grown in credibility over the years. Even before Wilesundertook to prove the Taniyama-Shimura conjecture, many mathematicians had come to believe that it isprobably true.

One reason the Taniyama-Shimuraconjecture seemed so unlikely at firstis that elliptic curves and modularforms are very different kinds of objects. To see how they are connected,consider again the set of rational pointson an elliptic curve. There are variousquestions one might ask about thesepoints for any given curve. How manyare there? If the number is finite, isthere a method for counting them? Arethere any patterns that govern wherethey appear along the curve? Can theybe classified?

A fruitful approach to such questionsis to think of the equation defining an el

liptic curve not as an equivalence but as acongruence modulo some prime number p. In other words, "reduce" the equation by dividing all values of x and y byp, saving only the remainder. We can illustrate this process for the elliptic curvedefined by the equation y2 + y = x3 - x2.The curve has just five rational points,namely (0, 0), (0, -1), (1, 0), (1, -1) andthe origin. Now consider the equationas a congruence modulo 7. All of thefive points listed above remain solutions of the equation modulo 7. What ismore, additional points, which do notlie on the curve itself, become solutionswhen the equation is reduced modulo7. For example, the point (5,1) becomesa solution because l2 + 1 modulo 7 iscongruent to 53 - 52 modulo 7 (theyboth leave a remainder of 2). Reducingmodulo 5 yields a different set of solution points, and reducing by 13 generates still another set.

In general, reduction is not possiblefor all primes. After the reduction, theequation must still specify a nonsingularcurve, which means mat the three rootsmust remain distinct modulo p. For anequation of the form y2 = x(x - A)(x + B),

with A and B meeting the various criteria set forth above, this condition is satisfied for all p that do not divide theproduct AB(A + B), or equivalently ABC.For the specific curve y2 = x(x - 3)(x + 32)the admissible primes are those thatdo not divide 3 x 32 x 35 = 3,360.Hence the curve cannot be reduced bythe primes 2, 3, 5 and 7, since they alldivide 3,360. The product of theprimes that divide ABC—in this case2x3x5x7 = 210—is called the conductor of the elliptic curve; it specifiesthe set of primes that give such "badreductions."

What is gained by reducing an elliptic curve modulo pi There is one immediate benefit: The number of rationalpoints on the reduced curve is guaranteed to be finite. Exploring a finite objectis often easier than studying an infiniteone. Beyond this consideration, there isthe hope that "local" solutions, relativeto a specific prime, may reveal something about the "global" solutions to theoriginal equation. In particular, one canstudy an elliptic curve by counting thenumber of solutions modulo p for manyprimes p (always excluding those that

1994 March-April 151

Figure 8. Repeated addition under the group law reveals the "order" of a point. Here, on the elliptic curve defined by the equation y2 + y =x3 + x2, the point (0, 0) is added to itself, then to the result of that operation, then to the next sum, and so on. The first addition (far left)yields the sum (1, -1), then adding (0, 0) to (1, -1) (second from left) yields (1, 0). The next sum (third from left) is (0, -1), but adding this

divide ABC). Observing how the number of points increases as p increases reveals information about the curve andespecially about the group law for therational points. The information is encoded in a mathematical object calledan L-series, made up from a recipe thatbegins with numbers designated a ,which measure how many points modulo p there are for each prime p. The exact relation between the L-series and thesize of the group of rational points is thesubject of an unproved conjecture; thebasic idea is that a curve with many rational points should have many pointsmodulo p for various p. The converseshould also be true.

The complete L-series is an infiniteproduct, incorporating information frominfinitely many primes, but finite approximations can be calculated to anydesired accuracy for any specific ellipticcurve by the direct method of countingrational points modulo many primes.This process is an arduous one, however,and the resulting L-series is in a formatmat makes it hard to use in further calculations. Modular forms offer a remarkable shortcut—at least for certain ellipticcurves, and possibly for all of them.

Modular forms come from a ratherdifferent realm of mathematics: Theyare analytic functions defined on thecomplex numbers. (A complex numberhas both a real and an imaginary part;the imaginary part is a multiple of /', thesquare root of -1.) Just as the real numbers can be arranged on a continuous

line, the complex numbers form a continuous plane, where every point hascoordinates written as x + iy. Modularforms are defined on the complex upper half-plane, the part of the complexplane with y > 0. In other words, amodular form is a function that takeseach complex number from the upperhalf-plane and returns as its value another complex number (or possibly thesame number).

A salient characteristic that distinguishes modular forms among thefunctions of complex analysis is thatthey are essentially invariant under certain transformations of the upper half-plane. These transformations, definedby square integer matrices, are knownas fractional-linear transformations. Forexample, each modular form/is invariant under integer translations: The value of/at the complex number z is thesame as its value at z + 1. In other casesthe modular form is not strictly invariant but is multiplied by a simple factor.The "level" of a modular form/is a positive integer that determines the set offractional-linear transformations thatleave/invariant. Roughly speaking, thespace of modular forms of level Ngrows as N grows. For example, thereare no nonzero modular forms of level12 or of level smaller than 10, but thereis a nonzero form of level 11; it is uniqueup to multiplication by a constant.

In connection with Fermat's last theorem and the Taniyama-Shimura conjecture, an important property of mod

ular forms is that they give rise to L-series analogous to those coming fromelliptic curves. Because modular formsand L-series both belong to the realmof complex analysis, studying the L-series attached to an elliptic curve becomes easier once it is known that thesame L-series is also attached to a modular form. And it is the content of theTaniyama-Shimura conjecture that foreach elliptic curve there is a modularform whose L-series is the same as thatof the elliptic curve.

Since elliptic curves are algebraic objects, the Taniyama-Shimura conjecturerepresents a deep connection betweenalgebra and complex analysis. Implausible as the conjecture seemed at first, itis now supported by a vast amount ofnumerical and philosophical evidence.Wiles's proof of the conjecture for amajor class of elliptic curves can beviewed as further support for this remarkable connection.

Proving E Is Not ModularThe sequence of events leading up toWiles's announcement last summerwas set in motion in 1985 with a conjecture by Gerhard Frey of the University of the Saarlands in Germany. Itwas Frey who drew attention to equations of the form y2 = x(x - A)(x + B),where A and B are supposed to comefrom a counterexample to Fermat'slast theorem. The corresponding elliptic curve is now often called the Freycurve. Because of constraints on the

152 American Scientist, Volume 82

origin (O) origin (O) origin (O)

point to (0, 0) (fourth from left) leads to the origin. Since the origin is the identity element, adding (0, 0) to the origin (fifth from left) necessarily returns the process to (0, 0) again. The point (0, 0) is said to be of order five, since five additions bring it back to the starting position. The sequence of operations is summarized in the diagram at far right.

numerical values of A and B, Freyunderstood that the elliptic curvey2 = x(x - A)(x + B) cannot be modular.He was unable to give a rigorousproof, but Jean-Pierre Serre of the College de France soon clarified preciselywhat was needed to justify Frey's insight: He formulated an explicit conjecture about modular forms whose truthwould imply that Frey was correct. Ayear later one of us (Ribet) supplied theproof. This result established a directlink between elliptic curves and Fermat's last theorem, since the Taniyama-Shimura conjecture asserts that nil elliptic curves are modular.

How can one demonstrate that an elliptic curve is or is not modular? For aspecific equation with known, numerical coefficients, there are computational methods of answering such questions; the methods are arduous butreliable. The computational methodscannot be employed in this instance,however, because the object of study isa curve whose existence is hypothetical. We can write down Frey's equationas y2 = x(x - A)(x + B), but unless weknow of a counterexample to Fermat'slast theorem, we cannot fill in numerical values for A and B; and if the theorem is true, of course, there are no suchvalues. Since there is no hope of calculating the properties of a curve we cannot exhibit, the only recourse is tochoose a less direct strategy.

The first stage in this process is toexamine a special subgroup of the

points on an elliptic curve. For a curveE and a chosen integer m, this subgroup is designated E[m], and it consists of those points of order dividingm; such points are called the ///-division points. Recall that the order of apoint is the number of times it must beadded to itself before the result isequal to the origin O. Thus the group£[///] is made up of all points that aftermultiplication by /// (or equivalentlyafter being added to themselves ///times) yield the origin. There is a reason for scrutinizing this rather curiousset of points. If the elliptic curve £ ismodular, then a study of £[///] revealsinformation about the modular formassociated with £. Moreover, there is asense in which £[///] itself can be modular, and if you can show that £[///] ismodular for an infinite number of ///,then you can show that E is modular.Conversely, if you can show that £[///]is not modular for some value of ///,then you know that E cannot be modular either.

A couple of caveats need to be mentioned here. First, on some ellipticcurves there are points of infinite order, which will not be included in theset £[///] for any finite value of ///. Second, the points in E[m] do not necessarily have integer coordinates oreven rational coordinates. The mostthat can be said of the coordinates isthat they are algebraic numbers: solutions of algebraic equations with rational coefficients.

How can one prove that £[///] is notmodular for some ///? The key is to set/// equal to //, the exponent in the hypothetical counterexample to Fermat'slast theorem. As noted above, in anysuch counterexample the product ABCis necessarily a perfect nth power. Butmaking ABC an /7th power gives thegroup E[n] some unusual properties,akin to those of the /7-division pointson an elliptic curve of conductor 2. It isalready known, however, that no elliptic curve of conductor 2 can possiblyexist; the smallest conductor is 11. Thesuggested connection with curves ofconductor 2 comes very near to beinga contradiction that would prove Fermat's last theorem directly, withoutreference to the Taniyama-Shimuraconjecture. Unfortunately, no one hasyet found a way to turn the hint into arigorous demonstration. The actualproof that E[n] is not modular followsa more circuitous path. On the hypothesis that E[//] is modular, it must be associated with a modular form of someminimal level. The essential point ofthe proof is to show that this level is 2,which is impossible since there are nononzero modular forms of level 2.

The argument supporting this conclusion is an intricate one, which wanders into still denser thickets of modernarithmetic. The place to begin is to lookat the action of a group of transformations, called a Galois group, on the elements of £[///], for each value of ///. Thedefinition of the Galois group will not

1994 March-April 153

be given here; it is enough to say thateach element of the group induces apermutation that "mixes up" the pointsin E[m] but nonetheless preserves theaddition law of points. In symbols, suppose that c> is an element of the Galoisgroup and P a point in £[///]: Then justas mP = O, so m(aP) = 0.

The permutations induced by the elements of the Galois group can be represented by 2 x 2 matrices whose entriesare integers modulo ;;/. The transformation arising from a is thus a matrix p(o). It is possible to view the pointsof £[///] as column vectors of integersmodulo m in such a way that the permutation induced by a becomes matrix multiplication by p(o). One saysthat the matrices form a representationof the Galois group. It is noteworthythat the representation preserves the

composition law of the Galois group;if o and x are elements of the groupthat combine to form a transformation\), then p(\)) is just the product of thematrices p(c) and p(x).

Let us pause to look back over ourpath so far. We began with the ellipticcurve £, defined by an equation inwhich the numbers A and B are supposed to come from a counterexampleto Fermat's last theorem. We turnedthen to the discrete set of points £[///],defined to include just those pointswhose order divides the integer ///. Weexamined how a certain Galois groupacts on £[///], and in particular lookedat the 2x2 matrices that form a representation of this group. Now at last wecan make a connection with the question of modularity. It turns out that thegroup of transformations we have

Figure 9. Some points of certain elliptic curves have infinite order: They can be added anunlimited number of times without ever reaching the origin. On the curve defined byi/2 = x(x - 3)(x + 32) all rational points except the three points on the x axis—(0, -32), (0, 0)and (0, 3)—are of infinite order. The diagram shows the first few points in the sequence ofpoints generated by repeated addition of (-4, 28). The first sum is (4, 12); then adding(-4, 28) to this point yields (-25, -70). The progression of sums can be continued indefinitely without ever reaching the origin and thereby entering a cycle.

reached through this tortuous argument supplies information on the elements of the L-series for the curve £.The existence of a formula for generating this series is one way of definingmodularity.

The connection between the L-seriesand the Galois group comes about asfollows. As noted above, the coefficients of the L-series are calculated byreducing the curve £ modulo p for various primes p. Each p yields one termin the series; specifically, the L-seriescoefficient a is equal to the differencebetween 1 + p and the number of rational points on £ reduced modulo p. Butnow the set £[///] of ///-division pointsprovides another way of interpretingap, at least for most prime numbers p.Specifically, for each p there is a distinguished element ap of the Galois groupand a corresponding matrix p(o\,) thatdepends on /// and p. The sum of thetwo diagonal elements of this matrix isa number modulo /// that is congruentto the integer ap.

The ability to recapture ap modulo ///via Galois theory is the starting pointof Ribet's proof that Frey's ellipticcurve £ cannot be modular. If youadopt the hypothesis that £ is indeedmodular, then the unusual propertiesof E[n] mentioned above allow you tofind a nonzero modular form of level 2that is related modulo // to the form associated with £. But there is no suchform of level 2, contradicting the original supposition that £ is modular. Inthis way the argument reaches its finalconclusion: If there is a counterexample to Fermat's last theorem, then theremust be at least one elliptic curve thatis not modular, contrary to the Taniyama-Shimura conjecture.

Proving £ Is ModularWiles has said that he began work on aproof of the Taniyama-Shimura conjecture as soon as he learned that it wouldimply a proof of Fermat's last theorem.His campaign to solve the problemwas to last seven years—and perhapsit has not ended yet.

Wiles's proof does not quite encompass the full Taniyama-Shimura conjecture; it excludes certain cases. When acurve is reduced modulo p, it is possible for all three roots to coalesce into asingle numerical value. An example isthe equation y2 = x(x - 10)(x + 15),where'the three roots 0,10 and -15 areall congruent to zero modulo 5. Wiles'sproof does not apply to curves of this

154 American Scientist, Volume 82

kind. It is restricted to semistable ellipticcurves, which are those with the property that whenever two of the roots coalesce modulo /;, the third root remainsdistinct. For an equation of the formi/2 = .v(.v - A)(x + B), the curve is semi-stable if no prime // divides both A andB. This condition clearly holds for theexample y2 = x(x - 3)(x + 32). In fact itholds for any equation derived from acounterexample to Fermat's last theorem, because of the congruences modulo 4 and modulo 32 imposed on Aand B. Hence the elliptic curve E derived from the counterexample mustbe semistable.

In setting out to prove that all semi-stable elliptic curves are modular,Wiles worked with the same mathematical tools employed in Ribet'sproof, and many more in addition—including some that did not yet existwhen Wiles began in 1986. Like Ribet,Wiles considers the set of points £[///]for which mP = O, and the resultingrepresentations of the Galois group.But in a crucial respect Wiles's task is aharder one. Where exhibiting a singlecounterexample was enough to complete Ribet's proof, Wiles must establish that £[///] is modular for an infiniteset of integers ///.

Wiles's basic strategy is to study thefamily of sets E[3], £[9], E[27], etc., or inother words the family defined by E[///v],where v is any positive integer. There isa good reason for choosing this particular series: In about 1980 Robert P. Lang-lands of the Institute for AdvancedStudy and Jerrold B. Tunnell of RutgersUniversity proved that £[3] itself is modular (Langlands 1980; Tunnell 1981).What the Langlands-Tunnell theoremmeans is that for any elliptic curve £, theset of points of order 3 forms a groupthat has an associated modular form.The trick is to extend that result to theentire family of sets £[3V].

Wiles effects this extension via arguments that hinge on representations ofthe Galois group. But there is a furthercomplication, albeit a minor one. Forthe proof to work, the representationdefined by the 3-division points of Emust be irreducible, in the sense that itcannot be built out of smaller representations. Wiles sidesteps this problemthrough a clever tactic. He shows thatif E is semistable, then either the representation coming from the 3-divisionpoints of E is irreducible, or else therepresentation arising from the 5-divi-sion points of E is irreducible. He then

Figure 10. "Reducing" the equation of an elliptic curve modulo a prime number p yields anew set of points that can be regarded as solutions of the equation modulo p. For the equation y2 + y = x3 + x2 there are only five rational solutions: (0, 0), (1, 0), (0, -1), (1, -1) and theorigin (left diagram). When the equation is reduced modulo 7 (right diagram), five morepoints satisfy the resulting congruence (blue dots). The original five points also remain solutions (red dots), although (0, -1) and (1, -1) are transformed into (0, 6) and (1, 6) by the modulo operation. Reduction modulo p is allowed only if the equation does not become singular.

Fermat equationa" + b" = cn

Elliptic curve Ey2 = x(x-a")(x+bn)

/ \E{p)

Rational pointsof E modulo p

#E(p)Number of rational

points on E modulo p

EMPoints whose

divides morder

Representations ofthe Galois group

ap=1+p-#E(p)pth coefficientof the L-series

\ 7

ap mod m = trace p(ap)pth coefficient modulo m

of the L-series for E

L-seriesencodes information

about rational points of E

Figure 11. L-series is a mathematical object that has a crucial role in the proof of Fermat's lasttheorem. One route to calculating the L-series (left branch of diagram) is to reduce the ellipticcurve E modulo p for many primes p; a formula relates the number of rational points on £ modulo /; to the /?th coefficient of the L-series. The other route (tight branch of diagram) examinesthe set Elm] of points whose order divides m for various integers m; a group of transformationsthat act on this set yields information modulo m about the coefficients of the L-series.

1994 March-Apr i l 155

supplies an elegant argument that allows him to work with the 5-divisionpoints when necessary, even though£[5] is not covered by the Langlands-Tunnell theorem.

At this point in the proof Wiles hasseveral collections of representations.Some of them come from modularforms and are therefore modular bydefinition. Others come from the elliptic curve £, and the aim is to prove thatthey are modular. It is possible to linkup the various collections by means ofa technique called deformation theory,introduced by Barry Mazur of HarvardUniversity. For this scheme to work,Wiles must show that every "deformation" of a representation p that mightplausibly be modular really is modular.His approach is based on counting: Heendeavors to show that there are nomore deformations than there aremodular forms. This is the most difficult and technical part of the proof. Itentails calculating an upper bound onthe size of an object called a Selmergroup. And it is in this part of the proofthat weaknesses have been reported.

Wiles announced his result at theend of a series of three lectures delivered at the new Isaac Newton Institutefor Mathematical Sciences at the University of Cambridge. Having statedhis main theorem—the Taniyama-Shimura conjecture applied to semi-stable elliptic curves—he added acorollary: If a" + b" = c", then abc = 0. Itseemed only appropriate that Fermat'slast theorem be treated as an incidental

afterthought, as something to be mentioned in passing—as Fermat himselfhad treated it 350 years earlier.

Wiles's proof (if it is confirmed) is apositive, constructive result. If the proofhad pertained only to Fermat's last theorem, it would have been a purely negative statement, denying the existenceof certain integers (those satisfying theFermat equation a" + b" = c"). But byproving a part of the Taniyama-Shimura conjecture, Wiles has also established that certain objects do exist—namely modular forms associated withall semistable elliptic curves. For example, Wiles's result implies that theequation y2 = x(x - 3)(x + 32) has suchan associated form.

Wiles's proof is written out in a 200-page manuscript submitted to the journal hwentiones Mathematicae. The paperhas five chapters, each of which hasample content to stand up as a journalarticle on its own. Most of the major results in arithmetic algebraic geometryof the past 25 years are cited.

The manuscript was sent to half adozen referees soon after the Cambridge talks but was not released forany wider circulation. What did circulate widely were rumors of trouble withthe proof, and last December Wilesposted a note of explanation in theUsenet news group sci.math: "Duringthe review process a number of problems emerged, most of which have beenresolved, but one in particular I havenot yet settled. The key reduction of(most cases of) the Taniyama-Shimura

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conjecture to the calculation of theSelmer group is correct. However the final calculation of a precise upper boundfor the Selmer group... is not yet complete as it stands. I believe that I will beable to finish this in the near future using the ideas explained in my Cambridge lectures." In February, whenWiles began a series of lectures atPrinceton, the gap had not been filled.

The setback is naturally disappointing, but there is every reason to sharein Wiles's optimism. Moreover, even inthe worst-case scenario, if the flaw cannot be repaired, the line of inquiry issurely not exhausted. There is more yetto come from modern arithmetic.BibliographyBell, E. T. 1961,1990. The Last Problem. Intro

duction and notes by Underwood Dudley.Washington, D.C.: Mathematical Association of America.

Buhler, J., R. Crandall, R. Ernvall and T. Met-sankyla. 1993. Irregular primes and cyclo-tomic invariants to four million. Mathematicsof Computation 61:151-153.

Cox, David A. 1994. Introduction to Fermat'slast theorem. American Mathematical Monthly101(1):3-14.

Edwards, Harold M. 1977. Fermat's List Theorem: A Genetic Introduction to Algebraic Number Theory. New York: Springer-Verlag.

Husemoller, Dale. 1987. Elliptic Curves. With anappendix by Ruth Lawrence. New York:Springer-Verlag.

Knapp, Anthony W. 1992. Elliptic Curves.Princeton: Princeton University Press.

Langlands, R. P. 1980. Base Change for GUI).Annals of Mathematics Studies Vol. 96.Princeton: Princeton University Press.

Ribenboim, Paulo. 1979.13 Lectures on Fermat'sLast Theorem. New York: Springer-Verlag.

Ribet, Kenneth A. 1990a. On modular representations of Gal(Q/Q) arising from modularforms, hwentiones Mathematicae 100:431̂ 76.

Ribet, Kenneth A. 1990b. From the Taniyama-Shimura conjecture to Fermat's last theorem. Annates de la Faculte des Sciences de1'Universitc de Toulouse 11:116-139.

Ribet, Kenneth A. 1993. Modular Elliptic Curvesand Fermat's Last Theorem. Selected Lecturesin Mathematics. Providence, R.I.: AmericanMathematical Society. Video recording.

Rubin, Karl, and Alice Silverberg. 1994. A reporton Wiles' Cambridge lectures. Bulletin of theAmerican Mathematical Society, to appear.

Silverman, Joseph H. 1986. The Arithmetic of Elliptic Curves. New York: Springer-Verlag.

Silverman, Joseph H., and John H. Tate. 1989,1992. Rational Points on Elliptic Curves. NewYork: Springer-Verlag.

Silverman, Joseph H., and Paul van Mulbregt.1992. Elliptic curve calculator: A collection ofMathematica routines to perform calculationson elliptic curves defined over the rationalnumbers. Available from the Internet mathematics archive at wuarcliive.wustl.edu.

Tunnell, J. 1981. Artin's conjecture for representations of octahedral type. Bulletin of theAmerican Mathematical Society 5:173-175.

156 American Scientist, Volume 82

Errata and AddendumIn the article "Fermat's Last Theorem and

Modern Arithmetic" (March-April) certainconstraints imposed on the equation A + B = Cwere expressed incorrectly. On page 148 thearticle states that A = 3 modulo 4, C = 1 modulo 4 and B is divisible by 32. These conditions are incompatible. They should havebeen applied to the symmetrical variant ofthe equation, /4 + B + C = 0,in which C is replaced by -C. (The replacement is permissiblebecause C represents c" in Fermat's equation,and n is odd.) One of the variables is divisibleby 32; since all three variables play identicalroles in the equation, no generality is lost bysupposing that this variable is B. Exactly oneof the remaining variables is 3 modulo 4, andone can suppose that this variable is A. Theremaining variable, C, is 1 modulo 4.

hi the article "The Predatory Behavior ofthe White Shark" (March-April) the number of white-shark attacks analyzed onvideotape recordings should have been 129.

The authors of the article "Behaviorism,Congitivism and the Neuropsychology ofMemory" (January-February), Herbert L.Petri and Mortimer Mishkin, wish to credit the seminal work on instrumental conditioning by Ruth W. Colwill (see especially material in The Psychology of Learning andMotivation, Academic Press, 1986).