Quaternion-Based Reformulation and Proof of Fermat’s Last Theorem and Its Link to Einstein’s Mass-Energy Relation in Hypercomplex Discrete Spacetime

We present a novel quaternion-based algebra framework to reformulate and elegantly prove 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 draw a physical analogy with Einstein’s mass–energy relation for quantized energy, momentum, and mass, which corresponds to the n=2n = 2n=2 case. For higher even exponents, the lack of integer solutions suggests a deeper constraint on discrete spacetime variables, motivating extensions to octonionic and sedenionic algebraic structures.