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 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 present a classical proof of Fermat’s Last Theorem for all exponents n ≥ 3. The argument reduces the general case to an odd prime exponent and then applies a structural result—Barlow’s Relations—together with p-adic valuation techniques. These tools force any hypothetical solution to satisfy rigid algebraic and prime-divisor constraints that are mutually incompatible. The contradiction holds uniformly in all cases, thereby eliminating every possible solution. The proof relies solely on elementary number theory, factorization identities, and valuation arguments, offering a conceptually simple route to Fermat’s Last Theorem that remains close to the arithmetic framework available in Fermat’s time.