Fermat's Last Theorem

Abstract

This is a nice expository paper at an elementary level which describes the type of mathematics used inWiles’ proof of Fermat’s Last Theorem. The first section of the paper is a discussion about tessellations of the plane with several nice examples from medieval artifacts, and concludes by mentioning the classification of planar crystallographic groups. The next section is an elementary introduction to elliptic curves, which mentions the standard way of contructing new points with rational coefficients from old ones by the secant and tangent method, and discusses the “local method” of counting the points on such curves modulo p. The third section is a discussion of Poincar´e’s hyperbolic plane and concludes with a nice picture due to Escher. In the fourth section,the author introduces the modular forms, and states the Taniyama-Shimura conjecture. The last two sections of the paper relate to Fermat’s equation, mention Frey’s idea of associating an elliptic curve to a would-be solution of Fermat’s equation, Ribet’s result to the effect that such a curve would not satisfy the Taniyama-Shimura conjecture, and finallyWiles’ proof of the fact that certain elliptic curves, including the particular Frey curve associated to a solution of Fermat’s equation, are modular. The paper is quite instructive for nonexperts, amateurs and college students alike.