A simple analytical proof of Fermat’s last theorem for n = 7

Abstract

It is well known that proof of Fermat’s last theorem for any odd prime is difficult and first proof for n  7 was given by Lame [1] ,and Kumar also has given a proof for a special class of primes (Regular primes)which includes the case n  7 .However, these proofs are lengthy and difficult and may not easily be extended for all odd primes. The prime n  7 differs from n  5 since 2.7.115 is not a prime, whereas 2.5111 is a prime. Then it follows from the famous theorem of Germain Sophie that the corresponding Fermat’s equation , ( , ) 1 7 7 7 z  x  y x y  may have two classes of integer solutions, xyz  0(mod7) and xyz  0(mod7) if we assume that the Fermat equation has non-trivial integer solutions for x, y, z . This fact is proved using the simple argument [3] of Oosterhuis. The main objective of this paper is to give a simple analytical proof for the Fermat’s last theorem n  7 using general respective parametric solutions corresponding two classes of solutions of the Fermat equation ,which has already been extended for all odd primes.