Abstract
has no integer solutions with X,Y, Z ≥ 1. Inspiring generations of work in number theory, its proof was finally achieved by Wiles. A qualitative result, Finite Fermat, was obtained earlier by Faltings; it says the Fermat equation has only a finite number of solutions (for each given n, up to rescaling). This paper is an appreciation of some of the topological intuitions behind number theory. It aims to trace a logical path from the classification of surface diffeomorphisms to the proof of Finite Fermat. The route we take is the following.
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