Integral points on a certain family of elliptic curves

TL;DR

This paper employs the Thue-Siegel method to bound the number of primitive integral solutions on a specific family of elliptic curves with j-invariant 1728, advancing understanding of their integral points.

Contribution

It introduces a new application of the Thue-Siegel method to bound integral points on elliptic curves with a fixed j-invariant, providing explicit upper bounds.

Findings

Established an upper bound for primitive integral solutions.

Bounded the number of integer points on the specified elliptic curves.

Demonstrated the effectiveness of the Thue-Siegel method in this context.

Abstract

The Thue-Siegel method is used to obtain an upper bound for the number of primitive integral solutions to a family of quartic Thue's inequalities. This will provide an upper bound for the number of integer points on a family of elliptic curves with j-invariant equal to 1728.