Proving Fermat’s Last Theorem Using Partial Differences of Powers and the Binomial Theorem

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 study seeks to revisit FLT through classical and elementary approaches that could have been accessible during Fermat’s era. Our work revisits FLT using classical methods, analyzing integers as differences of squares and oblongs and leveraging tools like partial differences and the Binomial Theorem. While identifying gaps in classical approaches, we propose generalized frameworks to extend the understanding of higher powers and integer sequences. The analysis reformulates the equation c^n = (y + h)^n and b^n = y^n for h > 0, offering a general framework for studying the difference of powers using elementary tools such as the Binomial Theorem and the Method of Common Differences. As a restricted case, we examine c = b + 1 and prove that the difference (b + 1)^n - b^n is never an n-th power for n > 2. This is extended to arbitrary differences h, demonstrating that (y + h)^n - y^n cannot equal a^n for any integer a > 1. The proof involves bounding arguments for large values of y, modular constraints, and telescoping sum representations. Additionally, the function M(n) https://oeis.org/A224996, which marks the transition point where differences of powers change behavior, is introduced as a key analytical tool. While this study relies on principles and methods accessible during Fermat’s time, it provides fresh insights into integer sequences, finite differences, and visual patterns in number theory. Beyond proving FLT for n > 2, the work contributes to a deeper understanding of the relationships between squares, oblong numbers, and higher powers, forming the foundation for future research in both classical and modern mathematics.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, integer sequences, historical proofs.2020 Mathematics Subject Classification: 11D41, 11A05, 11B37, 11Y70, 11B83.