Linear recurrence sequences without zeros

Dubickas, Artūras ; Novikas, Aivaras

Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 857-865

Résumé

Let $a_{d-1},\dots ,a_0 \in \mathbb Z$, where $d \in \mathbb N$ and $a_0 \neq 0$, and let $X=(x_n)_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}x_{n+d-1}+\dots +a_0x_{n}$ for $n=1,2,3,\dots $. We show that there are a prime number $p$ and $d$ integers $x_1,\dots ,x_d$ such that no element of the sequence $X=(x_n)_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p \geq 3$ and $d$ integers $x_1,\dots ,x_d$ such that every element of the sequence $X=(x_n)_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\dots ,p-s-1\}$.

Détail

DOI : 10.1007/s10587-014-0138-1

Classification : 11B37, 11B50, 11T06
Keywords: linear recurrence sequence; period modulo $p$; polynomial splitting in $\mathbb F_p[z]$