Perfect powers expressible as sums of two fifth or seventh powers

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.