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 novel quaternion algebraic framework to elegantly relate Fermat’s Last Theorem of an even power, without reliance on modular forms or elliptic curves. By embedding the Diophantine equation a2ⁿ + b2ⁿ = c2ⁿ into the complexified hypercomplex algebra ℍℂ, we define a noncommutative map A = an e₁ + bn e₂ + i cn e₃ in terms of three anti-commutative quaternion basis elements. Leveraging quaternionic exponential identities, we show that exp(i2pA) ≠ 1 for all 2n greater than 2, unless the integers a = b = c = 0, thus ruling out nontrivial solutions. We further note a physical analogy of the quadratic form in Einstein’s Pythagorean mass–energy relation for quantized energy, momentum, and mass, reflecting the case n = 2, while higher exponents lack integer solutions. This suggests a fundamental constraint on discrete spacetime variables, motivating extensions to higher-dimensional structures using octonions and sedenions.