HISTORY OF FERMAT'S LAST THEOREM

A THESIS

SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE

BY

BILL KALTSOUNIS

DEPARTMENT OF MATHEMATICS

ATLANTA, GEORGIA

AUGUST 1968 TABLE OF CONTENTS

Page

INTRODUCTION K K K ..>... . h K K K K K ... . 1

Chapter I. FERMAT'S LAST THEOREM 2

II. EARLY ATTEMPTS AT PROOF 8

III. RECENT WORK ON FERMAT'S LAST THEOREM .... 17

IV. SUMMARY 22

LIST OF REFERENCES 23

iii INTRODUCTION

The famed "Fermat's last theorem" has been chosen as a topic for

rather detailed study in order to provide a relatively brief summary

of the history and the research that has been done on the theorem as a possible stimulation and an aid to those who may be interested in

broadening the horizon of existing proof. Further research on the proof of the theorem or any of its innumer¬

able restricted cases is clearly dependent upon an almost complete know¬ ledge of existing research. Therefore a brief study of this sort should serve well as an introduction to the necessary knowledge and might possibly also be a link in a chain of work resulting in some new

ideas in the mathematical world. Since three centuries of research using known mathematical methods have resulted in no general proof, new methods or ideas must in all probability be developed before a general proof can evolve.

In order to provide an adequate introduction to the knowledge

necessary for further research, this study should include a brief his¬ tory of the birth of the theorem along with Fermat's actual and prob¬ able work on it, the plausibility of its validity, the minimum restricted

cases necessary for general proof, and lastly, a summary of the actual valid work that has been and is being done with a consideration of mo¬

tives for continued interest such as prizes for complete proof.

1 CHAPTER I

FERMAT'S LAST THEOREM

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadratos, et generaliter nullam in in- finitam ultra quadratum potestam in duas ejusdem nominis fas est dividere, dujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.l Pierre de Fermat (1601-1665), a member of the provincial parliament of Toulouse, became interested in the theory of numbers through the second edition of Bachet's translation of the Arithmetica by Diophantus. Fermat had a habit of scribbling notes in the margin of his copy. Con¬ sequently, since there was so little space available, his custom was to note down some conclusions that he had reached, and to omit the steps leading to that conclusion. In spite of this incompleteness of written proof, the epoch-making notes of Fermat made this 1670 edition, which incidentally was carelessly printed and untrustworthy as regards the text, precious to the future of mathematics. Beside the eighth proposition of the second book of Diophantus— 2 To divide a square number into two other square numbers. —Fermat has written the note appearing at the beginning of this chapter in Latin. Although this has not in almost three centuries been proven to be true, it is well known as "Fermat's last theorem." Translated

^G. A. Miller, Historical Introduction to Mathematical Literature (New York: The Macmillan Company, 1916), pp, 114-117. 2 David Eugene Smith, A Source Book in Mathematics (New York: McGraw-Hill Book Company, Inc., 1929), p. 213.

2 into English, Fermat's last theorem and his accompanying note of assurance of proof reads:

To divide a cube into two other cubes, a fourth power, or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.3

The question of whether Fermat possessed a valid demonstration of his last problem will in all likelihood forever remain an enigma. Fermat undoubtedly had one of the most powerful minds ever applied in mathematical endeavour, and from his indications and the since proven validity of many of his notes in the same book, one has every reason to believe that he was able or at least thought he was able to prove the assertions that he included in the Diophantus notes. The margin being too small was an observation he also made in many other instances

On the other hand, he may have made a mistake, as in another case, ot where the conjecture that all Fermat numbers (Ffc=2^ +1) were primes, which he repeated in several letters, proved incorrect. Fermat's con- nt _ z jecture that all numbers (Ft 2 +1) were primes was disproved by L. Euler in 1732. Euler showed that the number F^=2^ +1=4,294,967,297 having 10 digits is composite and can be divided by 641, As of 1963 we knew 38 composite numbers F^, namely for k=5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 23, 36, 38, 39, 55, 58, 63, 73, 77, 81, 117

125, 144, 150, 207, 226, 228, 260, 267, 268, 284, 316, 452, 1945. These 38 composite numbers F^ include those for which we know the

prime factors (for example F^ and F^), those whose prime factors we do not know, but for which we know the decomposition into products of two

integers >1 (such a number is ^^9^5)» an<^ those for which we do not know even the decomposition as products of two integers^ 1, although we know that such a decomposition exists (F^, Fg, F^g and F.^), The fact that Fermat was challenging world mathematicians with mathematical propositions which he himself did not seem to understand,

is also demonstrated in a letter he had written to the famous mathema- 4 tician Martin Mersenne (1588-1648), In his correspondence to Mercenne,

Fermat outlined three propositions which he believed to be true, (No of the form 12k+\L is a of any number of the form

3n+l, no prime number of the form 10k+l is a divisor of any number of the form 5n+l, and no prime number of the form 10k-l is a divisor of any number of the form 5n+l). However, A, Schinzel proved that for each of these propositions there exist infinitely many prime numbers for which it is fftlse.

An easy demonstration of Fermat's last theorem is possible using the assumption that a number can be resolved into prime (complex) factors in one and only one way. Although this assumption has been made by some writers and possibly Fermat, it is not universally true.

The general consensus seems to be that in view of the numerous investi¬ gations of the problem for three centuries, from every conceivable angle, by first and second-rate mathematicians, by amateurs and dilettanti, the existence of a proof based on any methods one can reasonably assume Fermat could have mastered is very unlikely. Such methods would un¬ doubtedly have great consequences in other problems, but

Fermat mentioned them nowhere. Like so many of the other mathematicians who worked on the problem, including Kummer whose results were the most incisive of all, Fermat may have fallen into one of the many pitfalls of insufficient reasoning, Fermat did, however, give, and this is his only known proof of all his theorems, a proof of the impossibility of dividing a fourth power into two fourth powers. Since the case involving cubes was pre¬ sented repeatedly by him as a challenge problem to French and English mathematicians, and that he should propose a problem to which he could not himself give an answer seems unlikely, he probably had a similar proof of this case, 4 4 4 Fermat's proof of the theorem that the equation x +y =z has no solution in positive integers is based upon his "method of infinite descent," The basic ideas involved in this method, which is essentially the Well-Ordering Principle which maintains that every non-empty set of positive integers contains a smallest member, follow. 5 If a proposition P(n) is true for some positive integers, then there is a least positive integer for which P(n) is true. But suppose it can be shown that the assumed truth of P(n) always implies the truth of P(n') where n* is a posi¬ tive integer less than n. Then a contradiction has been reached and the proposition P(n) must be false,4 Theorem: There are no natural numbers x, y, z satisfying the equation 4, 4 4 x +y =z . 4 4 4 Proof: Assume x +y =z where x, y and z are all prime to each other. Further it may be assumed that all the quantities referred to are posi¬ tive. As all numbers are either odd or even, x is of the form 2m or 2 2 2 2mfl, where m is an integer. Hence x is of the form 4m or 4m +4mt-l, that is of the form 4m or 4mbl, so that a number of the form 4m)-2 or 4mf3 cannot be a square. Hence x and y cannot both be odd, for then the sum of their forth powers would be of the form 4mt-2, and this cannot be a square. Hence either x or y must be even, and as it is obviously

2 2 2 2 immaterial which one is, suppose it is y. Since (x^) + (y ) =z , it 222 222 22 follows from equation x +y =z that we must have x =a -b , y=2ab, z=a +b , 2 2 2 where a and b are prime to each other, and not both odd. From x =a -h 2 we see that a cannot be even, for then b would be odd and x would be 2 2 2 of the form 4mt-3, which is impossible. We have then x +b =a where b is even, a is odd and prime to b, so that no two of a, b, x have a 2 2 2 2 2 common factor. Hence it follows from equation x +y =z that x=p -q , 2 2 b=2pq, a=p +q , where p and q are prime to each other and not both odd. 2 2 2 2 From y =2ab, we have y =4pq(p +q ). Since p and q are prime to each 2 2 other, each of them is prime to p +q , and hence all three must be per- 2 2222 442 feet squares. Put then p=r , q=s , p +q =t , from which r +s =t , Now 4 4 the values of x, y, z in terms of r, s, t are given by x=r -s , y=2rst, 228448 4424 4 z=a +b =r +6r s +s , so that z y (r +s ) } t or t

^B. M. Stewart, Theory of Numbers (New York: The Macmillan Company, 1952), p. 95. 6

continued, so that an infinite number of positive integers t, t1 , t„. , . ^ 2 can be found such that t^yz, t^yt^. ... This proves the impossibi¬ lity of the case of n=4, the method of proof being known as the method of infinite descent. The proof that xn+yn=zn is impossible when n=4 depends on the 2 2 2 solution of the case x +y =z . The fact that Fermat was able to supply a solution to the equation xn+yn=zn when n=4 indicates that he was aware 2 2 2 of the solution to the specific equation x +y =z , which is stated and proved below. Theorem; A general solution (i.e., a solution in which ((x, y, z,)=l) 2 2 2 2 2 of x +y =z , y even, x\ 0, y\ 0, z \ 0 is given by x=a -b , y=2ab, and 2 2 z=a +b , where a and b are prime to each other and not both odd, and a S b \ 0. 2 2 2 Proof; Suppose that x +y =z , since (x,y,z)=l, also (y,z)=l, so that (z-y, z+y)=l or 2. But z is odd and y is even, so that (z-y, z+y)=l. 2 Hence, from the equation x =(z~y)(z+y) we deduce that z-y and z+y must be squares , since they are positive. Now if t and u are fixed integers of the same parity (both odd or both even), there ate integers a and b 2 2 such that t=a+b and u=a-b. Hence we can put z-y=(.a**b) , z+y=(a+b) which gives z=- (a-b)^ + (arfb)^ _= a 2±b 2, y=_ (arfb)^ - (a-b)^ = 2ab, 2 2 o 2 and x=(a-b) (a+b) = a -b . Since (z-x, z+x)=(2a^, 2b2) =2, we must choose a and b so that (a,b)=l. Since x is odd, arfb must be odd. Since y^ 0, a and b must have the same sign, and since x^ 0, |a|> jbj. Since the pairs a, b and -a, -b give the same solution, we can suppose that a> b> 0. In summary, although Fermat left so little in the way of proof of his last theorem, and no general demonstration of it has yet been given, there is no reason to doubt its truth. Of the numerous investigations of special cases which have been made, not one has produced a general solution. The plausibility of the theorem seems even more evident with the fact that the analogous theorem in terms of functions (If g(x) is a rational_function of x,^g(x)dx is integrable in terms of elementary functions - that is a finite combination of algebraic, trigonometric and exponential functions together with their inverses) is relatively 7 easily proven as is shown in Solved and Unsolved Problems in Number Theory by Daniel Shanks. CHAPTER II

EARLY ATTEMPTS AT PROOF

Any attempts by mathematicians to provide a general solution

for the equation xn+yn=zn must be initiated with the thought of pro¬ viding logical justifications to the following properties. 1) The values of x, y and z are integral. If the theorem is

true for all integral values of x, y, and z, the it is also true for

all rational values of these variables because of the implications of the converse of this statement. 2) The variables x, y, and z are prime to each other. Suppose

(x,y) = (s^l) in xn+yn=zn. Then s divides z and x=sX, y=sY, and z=sZ

in which X, Y, and Z are prime to each other. 3) Either nM or n=p, an odd prime. If n=4 or p, then n=4k _ or pk for som k} 1. But xn+yn=zn is then impossible in integra} so¬

lutions of x, y, and z if (xk)^+(yk)^=(zk)^ an(j (xk)P+(yk)P=(zk)P are impossible. Although all known methods used in attempts at proof for

the restricted case n=p, an odd prime, begin by assuming that this case is satisfied in integers and then investigate their properties, the condition that this case is satisfied may also be expressed in

the form that there exists a rational r for which (l+r*V^^ is rational. The case n=4 and thus n=4k is the only case of the above, aside from several specific and restricted p*s, which has been proven to

date, and that by Fermat, as was demonstrated in Chapter I. _

Countless specific and restricted proofs have been made by pro¬ minent mathematicians since the time of Fermat, but never have these restricted proofs, even in combination with each other, reached the

breadth of the restrictions necessary for the general proof. Such renowned names in the mathematical world as Euler, Legendre, Dirichlet, Lam/, Germain, Mirimanoff, Kummer, and others appear in various ac¬ counts of the work done on the theorem as having contributed in some 8 9 way toward proof. Euler led the procession of proofs for specific values of n with his proof for the cubic, which was first published in a French trans¬ lation of his Algebra in 1770. Euler*s proof is similar to that of Fermat for n=4 depending upon showing that, if three integral values of x, y, and z can be found which satisfy the equation, then it will be possible to find three other and smaller integers which also sa¬ tisfy it, and in this way finally show that the equation must be satisfied by three values which obviously do not satisfy it. Thus, no integral solution is possible. This method, however, which has been applied so well to the cases already mentioned, seems to be inapplicable in any other case. Considering the limited number of restrictions given previously for a general proof, Euler's work along with Fermat's, established the validity of the theorem not only for n=3 and n=4, but for all values of n being multiples of 3 and 4. Dissatisfaction with proof only for specific cases led Euler to a conjecture which if true would have Fermat's last theorem as a special case. Euler conjectured that an nth power can never equal the 3 3 3 sum of fewer than n nth powers. For instance, although x +y =z has 3 3 3 3 no solution in positive integers, x -by +z =u has infinitely many solutions. This conjecture remains, as does Fermat’s last theorem, only a conjecture, 3 3 3 The case n=3, that is the equation x +y =z had been known to the Arabian mathematicians nearly seven hundred years before the time of Fermat, and a faulty proof of the impossibility had been given by them. It is very probable that Fermat discovered this special case before he discovered the general theorem, for he had proposed as a problem "to find values of x, y, and z satisfying the equation," and had later declared it was impossible, Euler was the first to prove the theorem for this special case, but this proof was incomplete in respect of an assumption wherein lay the real difficulty of the question, and which contained the germ of the development of the theory of ideals which was to be applied so successfully by Kummer many years later. Euler's proof as given in his Algebra is substantially as follows. 3 3 3 Theorem; There are no solutions of x +y =z in rational integers, except the trivial solution in which one of x, y, z is zero. Proof; Two of the unknowns x, y, and z must be odd, and as any of the unknowns may be either positive or negative, there is no loss of generality in supposing that z is even, and that x and y are both odd. Hence we can write xfy=2p, x-y=2q so that x=pfq, y=p-q, and the ori- 2 2 3 ginal equation becomes 2p(p +3q )=z , Now p and q are prime to each other, and cannot both be odd, for then x and y would not be prime 3 to each other. Further p cannot be odd and q even, for then z would be divisible by 2 and not by 8, which is impossible. Hence p must be 2 2 even and q odd so that p +3q is odd. Hence as p and q are prime to 2 2 each other, 2p and p +3q are either prime to each other or have a common factor 3, In the first case p and hence z are both prime to 3, while in the Batter case they are both divisible by 3, 2 2 Let us consider the first case in detail. As 2p and p +3q are prime to each other, each must be a perfect cube, so that we can write 2 2 3 2 2 p +3q =r , Values of p, q, r can be found by taking r=m +3n , where m _ ^ and n are integers, and writing pfq ^3=(mbn \f-3) , By equating real 3 2 2 3 and imaginary parts p=m -9mn , q=3m n-3n , and if m and n are prime to each other and not both odd, and m is not divisible by 3, then p and q are prime to each other and p is not divisible by 3, But though this 2 2 3 method gives suitable values of p, q, r satisfying p +3q =r , it is by no means obvious that all the values of p, q, r can be found in this way, though as a matter of fact if is so in this particular case. If 2 2 3 the equation had been p +llq =r , all_the values of p and q would not be given by putting pfq Y -11 = (mfny-li)^. The removal of this dif¬ ficulty involves the study of the arithmetical theory of the binary quadratic form, or of numbers. Now since 2p is a cube, the values of m and n are such that 2m(nrt-3n) (ro-3n) is a perfect cube. But since q=3n(nri-n) (m-n) is odd, n is odd and m is even. Hence since m is prime to 3, no two of 2m, mt3n, m-3n can have a common factor; and since their product is a per- 3 3 feet cube each of them must be a cube. Put then m+-3n=a , m-3n=b , 2m=c^, so that by addition a^+b^=c"^. Hence z^=2p(p^+3q^)=a^b^c^(m^+3n^) 2 2 6 3 3 6 or z=abc(m +3n )=1/3 abc(a +a b +b ), so that as a and b cannot both be 11 unity, z is numerically greater than c, It follows then, just as in the case when n=4, that we should have an infinite sequence of numeri¬ cally decreasing integers, which is impossible. The same result follows in the second case when z is divisible by 3. Following Euler’s lead, the procession moved on with numerous proofs for specific n's and thus their infinite numbers of multiples^ About 1825 the case n=5 was proved independently by the German mathe¬ matician Lejeune Dirichlet and the French Legendre. Dirichlet dis¬ cussed this case to quite an extent in his earliest treatise, Mémoire sur l’impossibilité de quelques equations indéterminés du cinquième degré, which he followed in seven years with a proof for n=14. Finally, in 1840, Lamé and Lebesque both presented proofs for the case n=7. Meanwhile other types of restricted proofs were being made, the most well-known being that of the Parisian mathematician and philoso¬ pher Sophie Germain, alias M, Le Blanc due to the difficulties im¬ posed upon intelligent women of the day. Sophie Germain's theorem that the equation x^+yP=z^ has no solution in integers prime to p if p is an odd prime, and if q=2p+l is also a prime, was proven by her for p ^100. Using similar methods and modifications of the value of q, this restricted case was proven by Legendre for p^200, by E, T. Maillet for p^223, and by L. E. Dickson for p^7000. Sophie Germain's proof, which shows how far one can go with very elementary arguments, follows. Theorem: The equation x^+y^=z^ has no solution in integers prime to p if p is an odd prime, and if q=2p+l is also a prime. Proof: Assume a solution. Take x, y, and z prime to each other not destroying the generality of the proof for the same reason as that expressed in restriction 2) of the restrictions for a general proof listed at the beginning of this chapter. Since p is odd, x^+y^=z^ may be written symmetrically as R^+S^+T^=0 where x=R, y=S, and z=-T. Consider S^^C^= (-Rj[^. By the theorem stating that if a+b and n^O, then a-b divides an-bn, S+T divides S^+T^ and thus also (-R)^. Since the solution must be in integers prime to p, p does not divide R and therefore does not divide S+T, Then use the theorem which states that 12 if m=kn ^ with k prime and n positive, (a,b)=l, u=am, v=bm, and

uk - vk _ k-1 ^ k-2 ^ . k-2 . k-1 . . w= — -u + u v + ... +uv + v , then (u-v,w)=ls or k according as (a-b,k)=l or k respectively. Let n=l and thus m=l, cP I rpP /_ -p \P k=p, u=a=S, -v=-b=T, and w= "g"~f— = ~ g+T * Since (S+TjP^1 from above making (a-b,k)=l, (u-v,w)=l or (&4-T, "g+Ç~" )=1» Now

P C-R^ since (-R) = “gff~ (S+T), both mm mm mm and (S+T) must be perfect S+T

C-R)P_ pth powers. Write S+T=r and = sZ±jl = dp making -R=rd by S+T SH-T

p TP + RP substitution. Similarly, by symmetry, T+R=s=s ., •S = se T + R and R+S=tP, = fP, -T = tf. Then 2R=sP+tP-rP if S+T=rP,

R+T=sP, and R+S=t^ are solved simultaneously for 2R. Now by Euler's criterion,if q=2p+l does not divide R, S, or T, then RP, SP, TP all f ±l(mod q). But RP+SP+TP=0, making this impossible because the least positive value of RP+SP+TP would be 4, p being the smallest odd prime 3. On the other hand, q cannot divide two or three of R, S, and T since, they are prime to each other. Therefore q must divide exactly one of them. Let it be R, From 2R=sP+tP-rP it therefore follows similarly that q divides exactly one of r, s, and t, and by

T+R=sP and R+S=tP, it must be r. Then since q divides R, from eP(T+R)=TP+RP, ePTs(mod q) or eP^TP”'*'(mod q) by restricted cancella¬ tion since T and q are relatively prime to each other. Now since q does not divide S, neither does it divide e, so by Euler's Criterion, eP=TP”"'*' S+l(mod q). From S+T=rP and the preceding conclusion that q divides r, S=-T(mod q). Therefore, from dP= =SP”^ -SP~^T+...+TP'’\

^Euler's Criterion: If a is relatively prime to the odd prime, p, then a is a quadratic residue or a quadratic nonresidue (mod p), ac¬ cording to whether a^5”^^ = l(mod p) or a^P"^^ s -1 (mod p). 13

P 'QM 1 n. 1 and the last two results, d =pS^ =pT^ =+p(mod q). However, this is impossible since (p/q)=+l, and by Euler’s Criterion the theorem is proven. Euler's Criterion provided the basis for the proof supplied by

Sophie Germain. Therefore, its proof will be of a considerable assistance in interpreting the various stages that Sophie Germain had 4 4 4 to go through in order to arrive at the solution of x +y =z .

Euler's Criterion: If a is relatively prime to the odd prime, p, then a is a quadratic residue or a quadratic nonresidue (mod p),

according to whether =l(mod p) or ^ =;.l(mod p). Proof: Using Fermat's theorem which states that if p is a prime number and a is an integer, then a^=a(mod p), it is found that

(p_1)/2 (a(p-l)/2_1)(a(p-l)/2+1)SaP-l..1=o(mod p) and thus a =l or a^^ ^ = -l(mod p). If a is a quadratic residue (mod p), there exists 2 an integer x^ such that x^ = a(mod p). By Fermat's theorem,

(p-l)/2_ . 2. (p-l)/2_ p-1., . . a — (XQ) =. XQ =l(mod p). To prove the converse, assume (p-1) /2 that a ^ sl(mod p). If g is a primitive root (mod p), there exists a positive integer t such that gfc5a(mod p). Then ^^=a^*>”'^^2=l(mod p) (The integer a is called a primitive root (mod p) if a belongs to the exponent k(m)(mod m)). Hence, according to a theorem which states that if at^l(mod m) and a belongs to the exponent k(mod m), then k|t, it is implied that t» 0(mod p-1). Thus, t is even and

t/2 t (§ ) 3. g =a(mod p) which further implies that a is a quadratic re¬ sidue (mod p),

A most significant step beyond the work of his predecessors was taken by the German mathematician E. E. Kummer in 1849. Rummer's attempt at proof was one of the two distinct sources, the other being

Gauss's law of biquadratic reciprocity, of the extension of the arith¬ metic of rational numbers to an arithmetic of algebraic numbers and considerably later to a partial arithmetization of linear algebra. 14 Algebraic numbers are numbers satisfying algebraicg equations with rational coefficients. By means of his "ideals" Kummer was able to

prove Fermat's last theorem for all multiples of primes which do not exceed 100 and for all the multiples of many larger primes.

Once at an early period Kummer thought that he had a complete proof of the theorem. He laid it before P. G. L. Dirichlet who pointed out that, although he had proven that any number

2+ c n** 1 f{a.)=c¥c^sà-c^a ••• n » where a is a complex nth root of unity, n is prime, and c, c^, c^, etc. are whole numbers, was the product of indecomposable factors, he had assumed that such a factorization was

unique, whereas this was not true in general. In other words, the

fundamental theorem of arithmetic does not hold in fields corresponding

to all primes. After years of study, Kummer concluded that this non¬ uniqueness of factorization was due to f(a) being too small a domain

of numbers to permit the presence in it of the true prime numbers. Undaunted by this totally unforeseen failure, Kummer created a

new kind of number, which he called "ideal," and the machinery of which,

says L. E, Dickson, is 'so delicate that an expert must handle it with the greatest care, and (is) nowadays chiefly of historical interest in

view of the simpler and more general theory of R, Dedekind, However, the ideas he introduced in this theory of ideals were of such impor¬

tance that no less an authority than Professor E. T. Bell is responsible for the statement that 'Rummer's introduction of ideals into arithmetic

was beyond all dispute one of the greatest mathematical advances of the g nineteenth century.' g We define an ideal R in D as a module in D with the special

property that if t€ DR then t^é^R. In symbols, i?o(^6 R, t£D then R(property valid for modules) (X t£R (property distinguishing ideals)

7 Florian Cajori, A History of Mathematics (New York: The Macmillan Company, 1919), pp, 442-443. g David Eugene Smith, A Source Book in Mathematics (New York: McGraw-Hill Book Company, Inc., 1929),pp. 119-126. 15 Using the theory of ideals and the definitions of a and a regular prime, Kummer proved Fermat’s last theorem to be true for every exponent which is a regular prime. The Bernoulli number B is a rational number defined by the power series 22 , B x2- ---— = l-x/2 + (-l)n —7““^ » • A prime p is regular if it * e -1 n=l 'WnV. divides none of the numerators of B^, B2, B^, B when these • • •» 2Zl 2 numbers are written in their lowest terms. Otherwise p is irregular. The only irregular primes of the first twenty-four odd primes, 2

— 3 x ' ^ nn i B^xr ascending powers of x, namely ----- = l-x/2 + >.(-l) ' so that e ~1 n-1 v ' B^=l/6, B2=l/30, B^l/42, ,,, are the well known Bernoulli’s numbers.

Then the required condition is that the numerators of one of the first 1/2(p-3) of the Bernoulli’s numbers should be divisible by p. The only primes less than 100 for which this condition is not satisfied are p=37, 59, 67, and hence it is proved that x3+y3=z3 is impossible if p is an odd prime less than 100, except when p=37, 59, 67,

In order to establish the truth of the theorem for these exceptional values of p, Kummer gave in 1857 some additional results for primes satisfying certain conditions. These conditions were satisfied by p=37,

59, 67, so that Fermat’s Last Theorem is proved for all values of p, prime or otherwise, less than 100, omitting of course p=2.

The first marked advance since Kummer, was made by A, Wieferich of

Munster and was published in Crelle’s Journal in 1909, He demonstrated the theorem that the equation x3+y3=z3 "has_.no solution in integers not 2 i>* X 2 divisible by p if p does not divide 2™ -1, or in other words, if p is not a "Wieferich square," A Wieferich square is the square of a prime 2 P“*l p such that p divides 2 -1, These squares are so rare that this criterion is sufficient for all p<100,000 except 1093 and 3511. Despite this rarity, no one has proven the conjecture that there are. infinitely 2 Vp-1 many p which satisfy the condition, p does not divide 2 -1, 2 Subsequently D, Mirimanoff showed in 1910 that p does not divide

iP-1 -1 is an equally valid criterion. CHAPTER III

RECENT WORK ON FERMAT'S LAST THEOREM

The fact that such eminent mathematicians as were mentioned in Chapter II were greatly interested in the proof of Fermat's last theorem was sufficient to secure for it considerable prominence in the mathematical world. On at least two occasions, 1850 and 1853, the "grand prix' des science mathématiques" (3000 francs) was offered for a complete solution by such an eminent body of scholars as the Paris Academy of Sciences. Although the secretary of this Academy received a considerable number of competing memoirs on both occasions, apparently none of them had enough merit to deserve the prize.

Stimulating much more interest than the prizes given by the Academy was a prize of 100,000 marks, an equivalent of nearly $25,000, bequeathed by Paul Wolfskehl in 1908 to the Konigliche Gesellschaft der Wissenchaften in Gottingen. This magnificent prize, the largest ever offered at that time for work in pure mathematics, was to be _ given either for a proof of the theorem or for a complete determina¬ tion of the values of n for which it is true, in case it is not uni¬ versally true. Such a vast amount of literature, most of which is as curious and as senseless as that of the circle squarers, resulted from the offering of the Fermat Prize as to lead to a special name, the literature of the Fermatists. More than a thousand false proofs were published during the first two or three years after the prize was announced. As a matter of fact, this theorem has the dubious distinc¬ tion of being the mathematical problem for which the greatest number of incorrect proofs have been published. Perhaps the seemingly elementary nature of the problem may explain the fact that so many 17 18 people without any mathematical insight or training have attempted solutions. This flood of false proofs, however, would seem to have been dammed somewhat by the rigorousness of the conditions under which the prize was to be awarded. Following are some of the conditions which were listed in Jahresbericht der Deutschen Mathematiker-Vereingung in June, 1908, No manuscripts will be considered, but only printed articles or monographs. These articles or mono¬ graphs must have appeared in regular periodicals, or they must have been for sale in the open market. The prize will not be awarded for articles or monographs which have been before the public for less than two years. It is suggested that five printed copies of the competing mémoires be sent by their authors to the said society at Gottingen, Germany,^ The competition for this prize is still open and will be until September 13, 2007, unless an acceptable solution is found before that time. Taking into consideration the German Inflation of the 1920’s and the apparent difficulties encountered in attempts at solu¬ tion even by some of the best mathematical minds in history, it would seem that there are easier ways to make money than by proving Fermat's 10 last theorem. Probably with no thought, or very little, of the still offered Fermat Prize, modern mathematicians continue to work on various cases of the theorem. Either intellectual curiosity or the urge to extend the boundaries of known mathematics must prolong the interest. Volumes of the Mathematical Reviews as recent as 1940-1962 reveal numerous reviews on various cases of Fermat's last theorem. As scientific technology has advanced, so have the lower bounds for the values of n been increased. Dr, J, Barkley Rosser has done a great deal of work along this line, Rosser considered only the first case of the theorem in which the equation x^+y^+z^=0 has no integral solutions x, y, and z in the of the pth roots of unity, with x, y, z prime to p, an odd prime. Employing the theorem of Morishima con-

le,, - * » . - . — — G, A, Miller, Historical Introduction to Mathematical Literature (New York: The Macmillan Company, 1916), pp. 114-117,

■^Daniel Shanks, Solved and Unsolved Problems in Number Theory (Washington, D, C.: Sparten Books, 1962), pp. 145-147 19 tending that if q is an improper prime, then for each prime 1 2 31, m^ ,=l(mod q ), Rosser proved, in 1939, that this first case held for all p<8,332,403 since there are no improper primes less than

8,332,403. Within the following year this lower bound for p was ex¬ tended to 41,000,000 by Rosser using the fact that q must exceed

8,000,000 to show that Mbrishima's theorem also holds for m=37 and m=41. Finally, using similar methods he extended the lower bound still further by extending m in the same theorem to 43, However, an indefinitely large bound does not seem possible by this method.

Dickson has proven this case for numbers up to 7,000 in 1908, and in

1925 Beeger pushed the bound up to 14,000. Using Rosser's results and extended theory, D. H. and Emma Lehmer, in 1941, raised the lower bound of the first case to all p<253,747,889.

Only twelve new cases of primes beyond this limit have been found, and those by K, Inkeri in 1959, Regarding the second restricted case, D. H. and Emma Lehmer along with Professor H, S. Vandiver have done quite a bit of work. The second case involves the same conditions as the first case excepting that p divides x. Desk calculator work first allowed Professor

Vandiver to validate the second case for all p<619, when calculations became prohibitively long. The job was then carried much further in only a few hours using the electronic calculator SWAC of the National

Bureau of Standards at Los Angeles under the supervision of J. Selfridge. Cooperation on the part of Vandiver, the Lehmers, and J. L. Nicol, combined with this use of the SWAC raised the limit to all p<2003, then in 1954 to all p<2521, and finally to all p<4002 in 1955. Not only has work been done on the exponential limits, but also on lower bounds for x and z in both cases, Inkeri showed in 1953 that in

Case I, x> while in Case II, x>p^^"^ and z>l/2p^ \

These inequalities lead to inconceivably large numbers when the known lower limits for p in the two cases are substituted. A partial reali-

An odd prime p is called improper if there are integers a, b, and c in the field of the pth roots of unity such that abc is prime to p and a^+bHc -0, 20

zation of the enormous size of these numbers might be gained from a

look at the same type of work done by L, Perez-Cacho on the lower

bounds of x and y. For p<1093 in Case I, y would have to be a number with at least 21,860 digits. Similarly in Case II, at least 14,000

digits would be required for y. Combining the two types of limits N. Raclis proved that if xn+yn=zn has no solution in positive integers with x

That n

both of the exponents and of x, y, and z, many different types of res¬ tricted proofs have been demonstrated. Some of these proofs are of

quite an elementary nature while others are of a nature too complicated

for mention here. The first case has intrigued several modern mathematicians, who

have arrived at varied results concerning p, M. Krasner, in 1940, proved the first case for all p such that there exists a prime q of u the form q=upfl, where u is not a multiple of 3, 3 r

using generally elementary methods of proof. In 1945 Vandiver, con¬

sidering the same q, gave a proof for u even and not divisible by 3,

p>u, and q>3 , Another restricted variation on Fermat’s last theorem which has enjoyed quite some popularity is the case in which the equation

x^+y^=z^, p an 0dd prime, has no solution in integers where p does not divide xyz. Kummer has been indirectly responsible for two of the recent results pertaining to the case under consideration. Franz

Niedermeier, in 1944, proved this case for none of x, y, or z being divisible by pq in which q is 3 or 5, p is a prime quadratic non- 12 residue of q. Six years later Max Gut showed that a connection 13 between the Fermat problem and the E exists much the

12 2 If the congruence x =a(mod m) has no solution, then a is called a quadratic non-residue (mod m).

13 An Euler number is a positive integer N such that every positive integer of the form ax^+by , where ab=N and ax is relatively prime to by, is either a prime number, the square of a prime number, the double of a prime, or a power of 2. There are sixty-five of these idoneal numbers. 21 14 same as the connection with the Bernoulli numbers indicated more than a century before by Rummer, He proved the case indicated at the beginning of this paragraph if at least one of Ep-5* Ep-7* E E ,, was divisible by p. p-9 p-11 More recently still, proofs for this case have been given which involve particular forms for p. In 1954 L, D. Grey demonstrated that p must be of the form 3n+l and that all odd prime factors of pfl must be of the form 4nrl-l, Louis Long, in 1960, found a simpler condition for the validity of the case in which p need only not be of the form

120nri-l or 120mf49. Two rather unrelated but possible useful results were found in the early 1950's, J, Izvekoff proved that the general form of Fermat's equation, xn+yn=zn with n>2 and x, y, and z integers, has no solution when x is a prime. Using quite elementary methods Mirko Mihaljinec later showed that the general form just mentioned cannot be solved in integers x, y, and z which are in arithmetical progression. Lastly, work is still being done on the simplification of already existing proofs for specific values of n. For example, a simple proof was given by F, J, Duarte in 1944 of the cubic, avoiding the usual discussion of the divisibility of x, y, and z by 2 and 3, The proof 3 is based on writing the equation in the form (xfy+z) = (xd-y) (y+z) (z+x) and then showing that it is possible to set x=3t+u and y=6t-u. Simple proofs have also been given by Robert Breusch for n=6 and n=10.

14The connection between the Fermat problem and the Bernoulli numbers was indicated on page 16 of this paper. CHAPTER IV

SUMMARY

The foregoing study of Fermat's last theorem has attempted to provide a brief compilation of the valid data existing on the theorem to serve as an introduction to those interested in broadening research. Included are a brief history of the theorem's beginnings along with existing valid research and the minimum restrictions necessary for general proof.

As can be discerned from the varied and recent dates of the work and results on Fermat's last theorem which have been mentioned, interest in Fermat's problem has remained remarkably active throughout its history, results and research appearing frequently in the mathematical journals.

Frankly admitted is the fact that the result, if ever obtained, would probably have little systematic significance for the general progress of mathematics. However, the indirect value of having the proof as a goal and a constant source of new efforts has been very significant, especially in the theory of algebraic numbers. Some of the new methods the theorem has inspired have proved to be basic not only for number theory, but also for many other branches of mathematics.

Rummer's beautiful theory of ideal numbers grew entirely out of his efforts to prove this theorem, and this theory is undoubtedly more important than a proof of Fermat's theorem based only upon methods known in his day would be.

Continual research on Fermat's last theorem expresses the hope that efforts to prove it may lead to other important new mathematical methods, and thus give rise to strong instruments for attacking other existing and new mathematical difficulties.

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24 25

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