Odd-Power Impossibility and a Constructive Path Toward Fermat's Last Theorem

Fermat could have soled FLT! We give a fully elementary constructive proof that for any odd prime p > 2 there are no non-trivial positive integer solutions to xp +yp =zp, xp−yp =zp, by reducing to the structured case ap ±bp = 2kpcp with a,b,c odd and pairwise co-prime, and applying parity, co-primality, and size-comparison arguments together with a single instance of the 2-adic Lifting-the-Exponent lemma. Combined with classical proofs for exponents 3 and 4, this yields a complete elementary proof of Fermat’s Last Theorem for all exponents n > 2.