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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 primitiveinteger solutions. Assuming GRH, we also prove the nonexistence of non-trivialprimitive integer solutions for the signatures (5,5,11), (5,5,13), and(7,7,11). The main ingredients for obtaining our results are descenttechniques, the method of Chabauty-Coleman, and the modular approach toDiophantine equations.

Perfect powers expressible as sums of two fifth or seventh powers