Elementary proof of Fermat's Last Theorem for even exponents

TL;DR

This paper provides an elementary proof of Fermat's Last Theorem specifically for even exponents by reducing the problem to the impossibility of extracting q-th roots from Pythagorean solutions.

Contribution

It offers a novel elementary proof for even exponents of Fermat's Last Theorem using properties of Pythagorean solutions, simplifying the traditional complex proof.

Findings

Fermat's Last Theorem holds for all even exponents.

Solutions to the Pythagorean equation cannot be q-th powers.

The proof extends the approach used for n=4 to all even n.

Abstract

A elementary proof of Fermat"s Last Theorem[1] is presented for the case of even exponents n=2q, where q is any integer, including 2. For even exponents, the proof of the theorem reduces to showing that solutions of the Pythagorean equation X_p,Y_p,Z_p are impossible to equate q-th powers X^q,Y^q,Z^q of Fermat"s equation solutions. In other words, Fermat"s equation with even exponents does not have a solution, due to the impossibility of extracting the q-th root from corresponding numbers X_p,Y_p,Z_p of the Pythagorean equation solutions. Similarly to Fermat"s proof for the case, n=4, the simplicity of the approach used here is based on the use of the Pythagorean equation solution.