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 showing the equation aⁿ + bⁿ = cⁿ has no positive integer solutions for exponents n greater than 2. This statement, now known as Fermat’s Last Theorem, remained unproven for centuries despite the efforts of countless mathematicians. Andrew Wiles’s work in 1994 finally provided a rigorous proof of Fermat’s Last Theorem. However, Wiles’s proof relied on advanced mathematical techniques far beyond the scope of Fermat’s time, raising questions about whether Fermat could have truly possessed a proof using only the methods available to him. Wiles’s achievement was widely celebrated, and he was awarded the Abel Prize in 2016; the citation described his proof as a “stunning advance” in mathematics. Combining short and elementary tools, we prove the Beal conjecture, a well-known generalization of Fermat’s Last Theorem. The present work potentially offers a solution closer in spirit to Fermat’s original idea.