Solving binomial Thue equations

TL;DR

This paper extensively computes solutions to binomial Thue equations of the form x^n - m y^n = ±1 for large bounds, using optimized algorithms and high-performance computing.

Contribution

It advances the solution of binomial Thue equations by performing large-scale computations for many values of m and specific exponents, extending previous results.

Findings

All solutions with max(|x|,|y|)<10^500 for m<10^7 and specified n are determined.

The optimized method of Pethő is effectively applied to large-scale computational problems.

The results provide comprehensive data on solutions to binomial Thue equations within the given bounds.

Abstract

We consider binomial Thue equations of type $x^n-my^n=\pm 1$ in $x,y\in Z$. Optimizing the method of Peth\H o we perform an extensive calculation by a high performance computer to determine all solutions with $\max(|x|,|y|)<10^{500}$ of binomial Thue equations for $m<10^7$ for exponents $n=3,4,5,7,11,13,17,19,23,29$.