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Representations of integers by powers of two primes

Mckenzie West and Natalie Wijesinghe

Vol. 16 (2023), No. 4, 563–577
Abstract

We examine certain representations of integers by prime powers. Specifically, we show that given two primes p and q, every integer has a bounded number of representations of the form pαqβ + pγ + qδ , with α, β, γ and δ nonnegative integers. Furthermore, we show that apart from finitely many integers and integers within a short list of infinite families, no positive integer allows more than one representation of this form.

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Keywords
$S$-units, powers of primes, Diophantine equations, Catalan conjecture
Mathematical Subject Classification
Primary: 11D85
Milestones
Received: 22 September 2021
Revised: 26 July 2022
Accepted: 3 September 2022
Published: 31 October 2023

Communicated by Bjorn Poonen
Authors
Mckenzie West
University of Wisconsin Eau Claire
Eau Claire, WI
United States
Natalie Wijesinghe
Department of Mathematics
Colorado State University
Fort Collins, CO
United States