AN ELEMENTARY PROOF OF FERMAT LAST THEOREM THEOPHILUS AGAMA Abstract. We provide an elementary proof of Fermat’s last theorem using the notion of olloids. 1. Introduction Fermat’s Last Theorem (FLT), one of the most famous and long-standing con- jectures in number theory, asserts that no three positive integers x, y, z can satisfy the equation x n + y n = z n for n ≥ 3. The conjecture, first proposed by Pierre de Fermat in 1637, was famously noted in the margin of his copy of an ancient Greek text, where he claimed to have discovered a ”marvelous proof” but lamented the narrow margin that prevented him from recording it. Despite Fermat’s assertion, no proof was found during his lifetime, and the conjecture remained unproven for over 350 years (Fermat, 1637). For much of this time, FLT was considered one of the most elusive conjectures in mathematics. Over the centuries, progress has been made by proving the theorem for specific values of the indices n. For example, Fermat himself proved the case for n = 4, and Euler demonstrated the result for n = 3 in the eighteenth century. Dirichlet and Legendre later proved the case for n = 5 in the 19th century (Fermat, 1637). Despite these partial results, the general case of FLT remained unsolved. A crucial breakthrough occurred in the 1990s with the work of Sir Andrew Wiles. After years of solitary work, Wiles presented his proof of FLT in 1994, utilizing advanced techniques from algebraic geometry, modular forms, and elliptic curves (Wiles, 1995). Central to Wiles’ approach was the Taniyama-Shimura-Weil conjec- ture (now a theorem), which posits a deep connection between elliptic curves and modular forms. Wiles’ proof hinged on showing that a particular type of elliptic curve could not exist, which in turn implied the truth of FLT. Although Wiles’ initial proof contained a gap, it was later corrected in 1995, solidifying the result and ending centuries of speculation on the validity of the theorem [1]. The work of several mathematicians played a critical role in the eventual proof. Ribets theorem in the 1980s was pivotal, showing that the truth of FLT was equiv- alent to a special case of the Taniyama-Shimura-Weil conjecture [2]. Ribet’s insight linked modular forms to the problem of Fermat’s Last Theorem, providing a crucial step toward the proof. The contributions of Frey, Nron, and others in the study of elliptic curves also formed the mathematical backdrop against which Wiles worked [4]. Moreover, the work of Pierre Deligne, whose proof of the Weil conjectures in the 1970s revolutionized algebraic geometry, laid the groundwork for understanding the Date : December 22, 2024. 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. olloids. 1