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In this paper we carry out the steps of Darmon's program for the generalizedFermat equation $$ x^p + y^p = z^5. $$ In particular, we develop the machinerynecessary to prove an optimal bound on the exponent $p$ for solutionssatisfying certain $2$-adic and $5$-adic conditions which are natural from thepoint of view of the method. The above equation is an example of a generalizedFermat equation for which higher dimensional Frey varieties are needed for themodular method to work and thus represents an interesting test case forDarmon's program.

A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves