Efficient resolution of Thue–Mahler equations

Gherga, Adela; Siksek, Samir

doi:10.2140/ant.2025.19.667

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Efficient resolution of Thue–Mahler equations

Adela Gherga and Samir Siksek

Vol. 19 (2025), No. 4, 667–714

DOI: 10.2140/ant.2025.19.667

Abstract

A Thue–Mahler equation is a Diophantine equation of the form

F (X, Y) = a \cdot p_{1}^{z_{1}} \dots p_{v}^{z_{v}}, \gcd (X, Y) = 1

where $F$ is an irreducible binary form of degree at least $3$ with integer coefficients, $a$ is a nonzero integer and $p_{1}, \dots, p_{v}$ are rational primes. Existing algorithms for resolving such equations require computations in the field $L = ℚ (𝜃, 𝜃^{'}, 𝜃^{''})$ , where $𝜃$ , $𝜃^{'}$ , $𝜃^{''}$ are distinct roots of $F (X, 1) = 0$ . We give a new algorithm that requires computations only in $K = ℚ (𝜃)$ making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell–Weil sieve that makes it practical to tackle Thue–Mahler equations of higher degree and with larger sets of primes than was previously possible. We give several examples including one of degree $11$ .

Let $P (m)$ denote the largest prime divisor of an integer $m \geq 2$ . As an application of our algorithm we determine all pairs $(X, Y)$ of coprime nonnegative integers such that $P (X^{4} - 2 Y^{4}) \leq 100$ , finding that there are precisely $49$ such pairs.

Keywords

Thue equation, Thue–Mahler equation, LLL, linear form in logarithms

Mathematical Subject Classification

Primary: 11D59

Secondary: 11D61

Milestones

Received: 28 July 2022

Revised: 9 April 2024

Accepted: 15 June 2024

Published: 24 March 2025

Authors

Adela Gherga
Tutte Institute for Mathematics and Computing Ottawa Ontario Canada

Samir Siksek
Mathematics Institute University of Warwick Coventry United Kingdom

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Vol. 19, No. 4, 2025