Abstract

This paper offers a comprehensive examination of Fermat's Last Theorem , a statement in number theory that captivated mathematicians for over 350 years until its proof by Andrew Wiles in 1994. Beginning with historical context surrounding Pierre de Fermat and the theorem's formulation , the paper meticulously reviews the mathematical foundations underlying the theorem, including Diophantine equations, modular forms, and elliptic curves. Special attention is given to Wiles' groundbreaking use of the Taniyama-Shimura-Weil conjecture and Ribet's theorem to provide a complete proof, including the resolution of an initial flaw in the proof. Furthermore, the paper explores the theorem's far-reaching implications in number theory, algebraic geometry, cryptography, and computer science. The study reveals that Fermat's Last Theorem is not just an isolated mathematical problem but a testament to the depth, beauty, and inter-connectedness of mathematics, with broad impact across various scientific disciplines.