Elementary proof of Fermat's Last Theorem for even exponents

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.