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\begin{document}

\title[The generalized Fermat equation $x^2+y^5=z^7$]{The generalized Fermat equation $\boldsymbol{x^2+y^5=z^7}$}
\author{Manvir Jaswal}
\address{Toronto, Canada}
\email{manvirjaswal@icloud.com}
\subjclass[2020]{Primary 11D41; Secondary 11R29, 11Y40, 11Y50}
\keywords{Generalized Fermat equation, Fermat--Catalan conjecture, Belyi map, fifth-power descent, power residue symbols, class number, explicit formula}
\hypersetup{pdftitle={The generalized Fermat equation x\unichar{"00B2}+y\unichar{"2075}=z\unichar{"2077}},pdfauthor={Manvir Jaswal}}

\begin{abstract}
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.
\end{abstract}

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\input{sections/03-descent}
\input{sections/04-local}
\input{sections/05-auxiliary}
\input{sections/06-selmer}
\input{sections/07-sieve}
\input{sections/08-control}

\appendix
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\input{sections/09-classnumber}
\input{sections/11-computations}

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