Representations of integers by powers of two primes

West, Mckenzie; Wijesinghe, Natalie

doi:10.2140/involve.2023.16.563

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Representations of integers by powers of two primes

Mckenzie West and Natalie Wijesinghe

Vol. 16 (2023), No. 4, 563–577

DOI: 10.2140/involve.2023.16.563

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 $p^{α} q^{β} + p^{γ} + q^{δ}$ , 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

Vol. 16, No. 4, 2023