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Keywords:
linear recurrence sequence; period modulo $p$; polynomial splitting in $\mathbb F_p[z]$
linear recurrence sequence; period modulo $p$; polynomial splitting in $\mathbb F_p[z]$
Summary:
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 $\{s+1,s+2,\dots ,p-s-1\}$.
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