@MISC{Merel_arithmeticof, author = {Loïc Merel}, title = {Arithmetic of Elliptic Curves and Diophantine Equations}, year = {} }

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Abstract

Introduction and background In 1952, P. Denes, from Budapest 1 , conjectured that three non-zero distinct n-th powers can not be in arithmetic progression when n > 2 [15], i.e. that the equation x n + y n = 2z n has no solution in integers x, y, z, n with x #= y, and n > 2. One cannot fail to notice that it is a variant of the Fermat-Wiles theorem. We would like to present the ideas which led H. Darmon and the author to the solution of Denes' problem in [13]. Many of them are those (due to Y. Hellegouarch, G. Frey, J.-P. Serre, B. Mazur, K. Ribet, A. Wiles, R. Taylor, ...) which led to the celebrated proof of Fermat's last theorem. Others originate in earlier work of Darmon (and Ribet). The proof of Fermat's last theorem