Perfect powers expressible as sums of two fifth or seventh powers
Tom 164 / 2014
Acta Arithmetica 164 (2014), 65-100 MSC: Primary 11D41; Secondary 11F80, 11G30. DOI: 10.4064/aa164-1-5
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 non-existence 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.