math.NT — Number Theory

The generalized Fermat equation $x^2+y^5=z^7$

Contributed by Manvir Jaswal

We show that the equation $x^2+y^5=z^7$ has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations $x^2+y^5=z^7$, $x^2+y^7=z^5$, $x^5+y^7=z^2$ has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree $8$, and proves that it is always isomorphic to one fixed octic field $L_8$. We prove unconditionally that no solution has fibre isomorphic to $L_8$, by a fifth-power descent over a field $L_{24}$ of degree $24$ that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that $L_{24}$ has class number $1$, using a criterion of Belabas, Diaz y Diaz and Friedman.