On the $p$-adic valuation of third order linear recurrence sequences

Abstract

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a generalisation of the conjecture of Marques and Lengyel for third order linear recurrence sequences. Suppose that $(x_n)$ is a third order linear recurrence sequence whose characteristic polynomial has a root $\gamma$ such that $|\gamma|>1$. We show that if there exists a prime $p$ for which the conjecture holds for $(x_n)$, then the solution set of the Diophantine equation given by $x_n=m!$ in positive integers $n,m$ is finite. We also show that the solutions can be effectively computed when the form of the conjecture is explicitly known.