Proof of Fermat’s Last Theorem of an Even Power Using Quaternion Algebra and the Link to Einstein’s Pythagorean Mass-Energy Relation in Discrete Spacetime

We present a new algebraic proof of Fermat’s Last Theorem (FLT) for all even exponents n = 2k > 2, based on an embedding of integer triples (a, b, c) into the complexified quaternion algebra ℍℂ with basis elements e₁, e₂, e₃. The method exploits the quaternionic exponential identity exp(i 2π A) = cos(2π‖A‖) + (iA/‖A‖) sin(2π‖A‖), where A is the quaternionic embedding of (a, b, c) and ‖A‖² = a2k + b2k − c2k. For integer solutions of an + bn = cn, the exponential condition exp(i 2π A) = 1 imposes strict trigonometric constraints that can only be satisfied by the trivial solution a = b = c = 0 when n > 2 and even. This approach avoids modular forms and elliptic curves, relying instead on noncommutative algebra and analytic properties of quaternion exponentials. The framework naturally extends to higher-dimensional Cayley–Dickson algebras, suggesting links between FLT-type problems, noncommutative geometry, and discrete hypercomplex models of spacetime.