nt.number theory - hard diophantine equation: $x^3 + y^5 = z^7$

I may be misunderstanding the question, but I do not believe that it has any integer solutions. At the very least, none are known to exist at the moment. Any solutions would be counterexamples to the Fermat-Catalan conjecture with {m,n,k} = {3,5,7} (since 1/3 + 1/5 + 1/7 = 71/105 < 1). The most I can tell you is that, for coprime {x,y,z}, there are finitely many solutions to your equation. I think your (x,y) = 1 means that they're coprime, anyway, so it follows that z must be coprime. Therefore, any solution at all would disprove the related Beal's conjecture.