Advanced Search

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