## Multiples of integral points on Mordell curves

### Volume 211 / 2023

Acta Arithmetica 211 (2023), 121-159
MSC: Primary 11G05.
DOI: 10.4064/aa220822-3-8
Published online: 16 October 2023

#### Abstract

Let $B$ be a sixth-power-free integer and $P$ be a non-torsion point on the Mordell curve $E_B:y^2=x^3+B$. We study the integral multiples $[n]P$ of $P$. Among other results, we show that $P$ has at most three integral multiples with $n \gt 1$. This result is sharp in the sense that there are points $P$ with exactly three integral multiples $[n]P$ and $n \gt 1$. As an application, we discuss the number of integral points on the quasi-minimal model of rank 1 Mordell curves.