ABSTRACT
In the late 1630s, Pierre de Fermat was reading Diophantus's Arithmetica, which had recently received a new translation from Greek into Latin by Bachet. The Arithmetica consists of hundreds of algebraic problems covering linear, quadratic, and higher degree equations and uses algebraic techniques to find their solutions. In the mid-nineteenth century, the great German mathematician Ernst Kummer began the development of modern algebraic number theory. Part of his work gave criteria that guarantee the validity of Fermat's Last Theorem for a given exponent n. Gerd Faltings's work made clear that the ideas of algebraic geometry have implications for Fermat's Last Theorem. Then in 1985, Gerhard Frey suggested that the Modularity Conjecture in the theory of elliptic curves could be used in proving Fermat's Last Theorem. Finally, in 1994, Andrew Wiles, with the assistance of Richard Taylor, proved the Modularity Conjecture and therefore completed the proof of Fermat's Last Theorem.
