Differences between perfect powers: prime power gaps

Bennett, Michael A.; Siksek, Samir

doi:10.2140/ant.2023.17.1789

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Differences between perfect powers: prime power gaps

Michael A. Bennett and Samir Siksek

Vol. 17 (2023), No. 10, 1789–1846

DOI: 10.2140/ant.2023.17.1789

Abstract

We develop machinery to explicitly determine, in many instances, when the difference $x^{2} - y^{n}$ is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and $S$ -unit equations, and the primitive divisor theorem of Bilu, Hanrot and Voutier) and classical algebraic number theory, with results derived from the modularity of Galois representations attached to Frey–Hellegoaurch elliptic curves. By way of example, we completely solve the equation

x^{2} + q^{α} = y^{n},

where $2 \leq q < 100$ is prime, and $x, y, α$ and $n$ are integers with $n \geq 3$ and $\gcd (x, y) = 1$ .

Keywords

exponential equation, Lucas sequence, shifted power, Galois representation, Frey curve, modularity, level lowering, Baker's bounds, Hilbert modular forms, Thue–Mahler equations

Mathematical Subject Classification

Primary: 11D61

Secondary: 11D41, 11F80

Milestones

Received: 11 October 2021

Revised: 22 September 2022

Accepted: 28 November 2022

Published: 19 September 2023

Authors

Michael A. Bennett
Department of Mathematics University of British Columbia Vancouver Canada

Samir Siksek
Mathematics Institute University of Warwick Coventry United Kingdom

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Vol. 17, No. 10, 2023