Cube sums of form $3p$ and $3p^2$

Jie Shu, Xu Song, Hongbo Yin

Let $p\equiv 2,5\mathrm{mod}9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also estab…