A Comprehensible Proof for Fermat's Last Theorem

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. We hypothesise that all r, s, and t are non-zero integers in the equation rp + sp = tp and establish a contradiction. To support the proof in the above equation, we have another equation: x³ + y³ = z³. Without loss of generality, we assume that both x and y are non-zero integers, z³ is a non-zero integer, and z and z² are irrational. We transform the above two equations using parameters, incorporating the Ramanujan-Nagell equation. Solving the transformed equations, we prove the theorem.