The generalized Fermat equation

Abstract

This article will be devoted to generalisations of Fermat’s equation xn + yn = zn. Very soon after the Wiles and Taylor proof of Fermat’s Last Theorem, it was wondered what would happen if the exponents in the three term equation would be chosen differently. Or if coefficients other than 1 would be chosen. We discuss the reduction of the resolution of such equations to the determination of rational points on finite sets of algebraic curves (over Q if possible) and explain the full resolution of the particular equation with exponents 2, 3, 5.