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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.
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
© 2023 MSP (Mathematical Sciences
Publishers).