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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-1-5,
author = {Sander R. Dahmen and Samir Siksek},
title = {Perfect powers expressible as sums of two fifth or seventh powers},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {65-100},
zbl = {1307.11041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-1-5}
}
Sander R. Dahmen; Samir Siksek. Perfect powers expressible as sums of two fifth or seventh powers. Acta Arithmetica, Tome 166 (2014) pp. 65-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-1-5/