Perfect powers expressible as sums of two fifth or seventh powers

2013/09/16 by Dahmen, Sander R., Siksek, Samir · 3 citations
#11D41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1309.4030

Abstract

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.

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