A Note on Fermat's Last Theorem

In 1637, Pierre de Fermat asserted that the equation aⁿ + bⁿ = cⁿ has no positive integer solutions for any exponent n > 2, famously claiming to possess a proof too large for the margin. Although Andrew Wiles established the full theorem in 1994 using deep methods from algebraic geometry and modular forms, the possibility of a more elementary argument has remained a topic of enduring interest. In this work we give a classical proof of the nonexistence of solutions to the Fermat equation for prime exponents under a natural local constraint: we assume that for an odd prime p, any hypothetical solution triple (a, b, c) to a^p + b^p = c^p satisfies a + b − c < 2p. The proof proceeds by establishing that the difference δ = a + b − c must satisfy δ ≥ 2p, thereby contradicting the hypothesis. This follows from three elementary observations: first, by the binomial theorem, (a + b)^p > a^p + b^p = c^p, ensuring δ > 0; second, by Fermat’s Little Theorem applied modulo p, we have p | δ, yielding δ ≥ p; third, parity considerations force 2 | δ, which combined with δ ≥ p gives δ ≥ 2p. The argument relies solely on elementary number theory and congruence techniques, remaining close to the arithmetic framework available in Fermat’s time.