An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R - P.N Seetharaman, (Indian Journal of Advanced Mathematics (IJAM), 2025)
In number theory, Fermat’s Last Theorem states that no three positive integers a, b and c satisfy the equation a n + b n = c n where n is any integer > 2. Fermat and Euler had already proved that there are no integral solutions to the equations x 3 + y3 = z3 and x4 + y4 = z4 . Hence it would suffice to prove the theorem for the index n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + sp = tp where p is any prime >3 and prove the theorem using the method of contradiction. We have used an Auxiliary equations x 3 + y3 = z3 along with the main equation rp + sp = tp , which are connected by means of transformation equation through the parameters. Solving the through transformation equations we get the result rst = 0, showing that only a trivial solution exists in the main equation.