### 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 2.7.115 is not a prime, whereas 2.5111 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.