In Search of the Truly Marvelous Proof

Abstract: Fermat's Last Theorem (FLT), proposed in 1637 by Pierre de Fermat, states that no positive integers a,b, and c satisfy the equation a^n+b^n=c^n for any integer n>2. While Fermat claimed to have discovered a "truly marvelous proof" for this result, his proof was never found, leading to centuries of attempts to verify the theorem. In 1994, Andrew Wiles produced the first rigorous proof using advanced techniques in algebraic geometry and modular forms. This work revisits FLT using classical methods: (i) binomial expansions, (ii) finite differences, (iii) modular arithmetic, and (iv) bounding arguments on the growth of consecutive powers. We focus on the impossibility of expressing (y+h)^n-y^n as another perfect n-th power, for n>2. Our analysis combines bounding arguments for large values of y and exhaustive modular checks for small values, supplemented by references to Baker’s theorem for further rigor in excluding integer solutions. Additionally, we introduce the function M(n,h) (https://oeis.org/A224996 on OEIS for h=1) as a key analytical tool that marks the transition points in the growth of these differences. While the methods employed here align with historical techniques available in Fermat’s time, they also reveal how such "elementary" routes, though rich in insight, encounter significant hurdles that modern number theory ultimately resolved via deeper structures. However, it remains conceivable—at least in spirit—that Fermat might have perceived a further purely modular "incoherence," never rediscovered, which would have sealed the proof without the machinery of the 20th century and beyond.This study, presented as a preprint, offers preliminary insights that are yet to undergo peer review. Future revisions may further refine the findings and conclusions drawn here.Keywords: Fermat's Last Theorem, number theory, partial differences, method of common differences, Binomial Theorem, polynomial growth, modular arithmetic, historical proofs.2020 Mathematics Subject Classification: 11D41, 11A05, 11B37, 11Y70, 11B83.