A trigonometric centered exploration for Fermat’s last theorem

In 1995, Andrew Wiles completed his landmark proof of Fermat’s Last Theorem, drawing upon deep machinery from algebraic geometry, modular forms, and the theory of elliptic curves, most notably the Taniyama–Shimura–Weil conjecture. While this achievement resolved one of the most celebrated problems in mathematics, the proof’s extraordinary sophistication and reliance on twentieth-century techniques have sustained a long-standing desire within the mathematical community for a more elementary or concise demonstration. A simpler solution, one that circumvents the heavy apparatus of modern arithmetic geometry and instead rests on classical or foundational principles would not only illuminate the intrinsic structure of the theorem but also provide greater conceptual transparency. Consequently, the search for alternative, more accessible proofs remains an active and compelling intellectual pursuit. In this paper a simple, novel trigonometric approach to Fermat’s Last Theorem is presented. The Fermat’s last theorem is first reduced to a trigonometric form and then it shown that it is not possible for the trigonometric form to exist.