On the p-Adic Valuation of Third Order Linear Recurrence Sequences

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 \((x_n)\) $(x_n)$ is a third order linear recurrence sequence whose characteristic polynomial has a root \(\gamma \) $\gamma$ such that \(|\gamma |>1\) $|\gamma |>1$ . We show that if there exists a prime p for which the conjecture holds for \((x_n)\) $(x_n)$ , then the solution set of the Diophantine equation given by \(x_n=m!\) $x_n=m!$ 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.