The generalized Fermat equation $x^3+y^3=z^n$
We prove that the equations $x^3+y^3=z^n$ and $x^3+y^3=3z^n$ have no solution in coprime nonzero integers for any $n\ge3$. For the first equation the open cases were prime exponents $p\equiv1\pmod 3$ above $10^9$ outside a set of congruence classes treated by Chen and Siksek; in these cases the solution $1^3+2^3=3^2$ blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over $K=\mathbb{Q}(\sqrt{-3})$, whose parameter at a solution is $3$-adically close enough to a point of maximally unipotent monodromy for inertia at $\sqrt{-3}$ to act unipotently;…
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