Pierre de Fermat

Pierre de FermatThe most tantalizing marginal note in the history of mathematics. Of the well over three thousand mathematical papers and notes that he wrote, Fermat published only one, and that just five years before his death and under the concealing initials M. P. E. A. S. Many of his mathematical findings were disclosed in letters to fellow mathematicians and in marginal notes inserted in his copy of Bachet's translation of Diophantus's Arithmetical At the side of Problem 8 of Book II in his copy of Diophantus, Fermat wrote what has become the most tantalizing marginal note in the history of mathematics. The considered problem in Diophantus is: " To divide a given square number into two squares." Fermat's accompanying marginal note reads: To divide a cube into two cubes, a fourth power, or in general any power whatever above the second, into two powers of the same denomination, is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it. This famous conjecture, which says that there do not exist positive integers x, y, z, n such that xn + yn = zn when n > 2, has become known as "Fermat's last theorem." Whether Fermat really possessed a sound demonstration of this conjecture will probably forever remain an enigma. Because of his unquestionable integrity we must accept as a fact that he thought he had a proof, and because of his paramount ability we must accept as a fact that if the proof contained a fallacy then that fallacy must have been very subtle. Many of the most prominent mathematicians since Fermat's time have tried their skill on the problem, but the general conjecture still remains open. There is a proof given elsewhere by Fermat for the case n = 4, and Euler supplied a proof (later perfected by others) for n = 3. About 1825, independent proofs for the case n = 5 were given by Legendre and Dirichlet, and in 1839 Lame proved the conjecture for n = 7. Very significant advances in the study of the problem were made by the German mathematician E. Kummer. In 1843, Kummer submitted a purported proof of the general conjecture to Dirichlet, who pointed out an error in the reasoning. Kummer then returned to the problem with renewed vigor, and a few years later, after developing an important allied subject in higher algebra called the theory of ideals, derived very general conditions for the insolvability of the Fermat relation. Almost all important subsequent progress on the problem has been based on Kummer's investigations. It is now known that " Fermat's last theorem " is certainly true for all n < 4003 (this was shown in 1955, with the aid of the SWAC digital computer), and for many other special values of n. In 1908, the German mathematician P. Wollskehl bequeathed 100,000 marks to the Academy of Science at Gottingen as a prize for the first complete proof of the "theorem." The result was a deluge of alleged proofs by glory-seeking and money-seeking laymen, and, ever since, the problem has haunted amateurs somewhat as does the trisection of an arbitrary angle and the squaring of the circle. " Fermat's last theorem" has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published.

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