Generalized Fermat equations: A miscellany

M.A. Bennett, I. Chen, S.R. Dahmen, S. Yazdani

Research output: Contribution to JournalArticle

Abstract

This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.
Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalInternational Journal of Number Theory
Volume11
Issue number1
Early online date8 Jul 2014
DOIs
Publication statusPublished - 2015

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