On the k-regularity of the k-adic valuation of Lucas sequences

Abstract

For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n + 1} + b u_n$, for some integers $a$ and $b$. Shu and Yao showed that for any prime number $p$ the sequence $v_p(u_{n + 1})$ is $p$-regular, while Medina and Rowland found the rank of $v_p(F_{n + 1})$, where $F_n$ is the $n$-th Fibonacci number. We prove that if $k$ and $b$ are relatively prime then $v_k(u_{n + 1})$ is a $k$-regular sequence, and for $k$ a prime number we also determine its rank. Furthermore, as an intermediate result, we give explicit formulas for $v_k(u_n)$, generalizing a previous theorem of Sanna concerning $p$-adic valuations of Lucas sequences.