Multiples of integral points on Mordell curves

TL;DR

This paper investigates the number of integral multiples of points on Mordell curves, establishing an upper bound of three such multiples for non-torsion points and exploring implications for integral points on related curves.

Contribution

It proves that non-torsion points on certain Mordell curves have at most three integral multiples, providing a sharp bound and analyzing implications for integral points on rank 1 curves.

Findings

Maximum of three integral multiples for non-torsion points

Existence of points with exactly three integral multiples

Implications for integral points on quasi-minimal models

Abstract

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