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Differences between perfect powers: prime power gaps

Michael A. Bennett and Samir Siksek

Vol. 17 (2023), No. 10, 1789–1846
Abstract

We develop machinery to explicitly determine, in many instances, when the difference x2 yn 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

x2 + qα = yn,

where 2 q < 100 is prime, and x,y,α and n are integers with n 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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