Sylvester’s problem and mock Heegner points

Dasgupta, Samit; Voight, John

doi:10.1090/proc/14008

Sylvester’s problem and mock Heegner points
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by Samit Dasgupta and John Voight PDF

Proc. Amer. Math. Soc. 146 (2018), 3257-3273 Request permission

Abstract:

We prove that if $p \equiv 4,7 \pmod {9}$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y \in \mathbb {Q}$.

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