A Brief Elementary Proof for Fermat’s Last Theorem Using Ramanujan-Nagell Equation

Abstract

Fermat's Last Theorem states that it is impossible to find natural numbers A, B and C satisfying the equation (where is any integer ). Fermat himself proved the theorem for the index and Euler proved for [1]. In the equation we hypothesize that all r, s and are non-zero integers, and prove the theorem by the method of contradiction. Merly for supporting the proof in the above equation, we include another equation without loss of generality, we assert that both and as non-zero integers; a non-zero integer; and irrational. By trial and error method, we have created transformation equations to the above two equations, into which we have incorporated the Ramanujan-Nagell Equation and on solving the two transformation equations with the aid of Ramanujan--Nagell Equation, we prove the theorem by showing