Simple proof of Fermat’s last theorem for n =11

Abstract

Proof of Fermat’s last theorem for any odd prime is difficult. It may be extremely difficult to generalize any available Proof of Fermat’s last theorem for small prime such as n  3,5,7 to n 11[1]. The prime n 11 is different from n 13,17,19 in the sense that 2n 1 23 is also a prime and hence the corresponding Fermat equation may have only one type (Class.2) of solutions due to a theorem of Germaine Sophie[1],[2]. In this contribution, we will give a simple proof for the exponent n 11 based on elementary mathematics. The Darbrusow identity[1] that we will use in the proof can be obtained as Darbrusow did using the multinomial theorem on three components[1]. In our proof, it is assumed that the Fermat equation 11 11 11 z  y  x , (x, y) 1 has non-trivial integer solutions for (x, y, z) and the parametric solution of the equation is obtained using elementary mathematics. The proof of the theorem is done by showing that the necessary condition that must be satisfied by the parameters is never satisfied.