On products of prime powers in linear recurrence sequences
Volume 215 / 2024
Acta Arithmetica 215 (2024), 355-384
MSC: Primary 11D45; Secondary 11B37, 11D61
DOI: 10.4064/aa230911-22-4
Published online: 16 September 2024
Abstract
We consider the Diophantine equation $U_n=p^xq^y$, where $U=(U_n)_{n\geq 0}$ is a linear recurrence sequence, $p$ and $q$ are distinct prime numbers and $x,y$ are non-negative integers not both zero. We show that under some technical assumptions the Diophantine equation $U_n=p^xq^y$ has at most two solutions $(n,x,y)$ provided that $p,q\notin S$, where $S$ is a finite, effectively computable set of primes, depending only on $U$.
