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History of Fermat's Last Theorem a Thesis Submitted to the Faculty of Atlanta University in Partial Fulfillment of the Requireme

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History of Fermat's Last Theorem a Thesis Submitted to the Faculty of Atlanta University in Partial Fulfillment of the Requireme 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 conjecture 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 prime number of the form 12k+\L is a divisor 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 number theory 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.
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