Generalized Fermat equations with three distinct prime exponents

We consider the problem of solving $x^p+y^q=z^r$ in nonzero
coprime integers x,y,z in the notorious case when the exponent triple
(p,q,r) consists of three distinct primes other than (2,3,q). We focus
in particular on the cases (p,q,r)=(2,5,7) or (3,5,7), where partial
resolutions can be obtained, i.e. solutions under stringent congruence
conditions. The techniques used involve Hunter searching of number
fields aided by computing p-adic étale algebras, and determining
rational points on curves. This contains joint work with Samir Siksek
and Casper Putz.