A Note on Fermat's Last Theorem

Around 1637, Pierre de Fermat famously wrote in the margin of a book that he had a proof for the equation aⁿ + bⁿ = cⁿ having no positive integer solutions for exponents n > 2. While Andrew Wiles provided a complete proof in 1994 using advanced 20th-century machinery, the question of whether a simpler proof exists remains a subject of intense mathematical interest. In this work, we focus on a significant restricted case of the theorem: the situation in which the exponent n possesses a prime divisor p that does not divide the quantity abc. Under this natural arithmetic condition, we develop an elementary argument—based on Barlow’s Relations and p-adic valuations—that leads to a contradiction. These methods lie closer to the classical number-theoretic framework that Fermat himself might have envisioned, and they illuminate structural features of the Fermat equation that persist across related Diophantine problems.