Cubic Thue Equations

TL;DR

This paper proves that for large discriminant cubic binary forms, the Thue equation |F(x, y)| = 1 has at most 7 integer solutions, refining understanding of solutions to these classical Diophantine equations.

Contribution

It establishes an upper bound of 7 solutions for large discriminant cubic Thue equations, improving previous bounds and revisiting earlier work by R. Okazaki.

Findings

At most 7 solutions for large discriminant cubic Thue equations

Refinement of bounds on solutions to cubic Thue equations

Extension of Okazaki's previous results

Abstract

We revisit a work by R. Okazaki and prove that for every cubic binary form F(x, y) with large enough discriminant, the Thue equation |F(x, y)| = 1 has at most 7 solutions in integers x and y.