Notes 101.01 Fermat's last theorem for n = 3 revisited

摘要

In the 17th century, Fermat discovered the insolvability in non-zero integers of the equation x3 + y3 = z3 (but his proof has been lost). In 1770, Euler gave a proof (see [1, pp. 39-42]), followed by Gauss (see [1, pp. 42- 45]) and Legendre (see [2, pp. 357-360]). Various proofs were obtained by Lamé, Tait, Günther, Gambioli, Krey, Thue, Carmichael and by many other mathematicians. A proof in Euler's style is based on a crucial lemma, followed by an infinite descent that is divided in two main cases (according as to whether the g.c.d. of x +y and x2 - xy + y2 is 1 or 3).In the present version, the crucial lemma receives a short proof and an additional observation allows elimination of one of the two main cases.