A Solution for Fermat’s Last Theorem Using Ramanujan-Nagell Equation

Abstract

Fermat’s Last Theorem states that it is impossible to find positive integers A, B, and C satisfying the equation: An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3 [1]. We hypothesize that all r, s and t are non-zero integers in the equation: rp + sp = tp and establish contradiction. Just to support the proof in the above equation, we have another equation: x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 is a non-zero integer; and z and z2 are irrational. We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan - Nagell equation. Solving the transformed equations, we prove the theorem.