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Linear recurrence sequences without zeros. (English) Zbl 1349.11017

There are two main results in the paper. In order to quote them we recall the following. Given a positive integer \(d\) and \(2d\) fixed integers \(a_0,\dots ,a_{d-1}\) and \(x_0,\dots ,x_{d-1}\) with nonzero \(a_0\); a linear recurrent sequence \((x_n)\) of order \(d\) is defined by these data if \[ x_{n+d}=\sum_{j=0}^{d-1}a_jx_{n+j} \] for all positive integers \(n\). Call \(char(F,t)\) the monic polynomial of degree \(d\) with coefficient \(-a_j\) in \(t^j.\) For a polynomial \(A(t)\) let \(P(A)\) be the set of all prime numbers \(p\) for which \(A(t)\) has a nonzero root in \(\mathbb{Z}/p\mathbb{Z}\).
Theorem 1. There are a prime number \(p\) and \(d\) integers \(x_0,\dots ,x_{d-1}\) such that no element of the sequence \((x_n)\) is divisible by \(p\). Moreover, \(p\) can be taken in \(P(char(F,t))\).
Theorem 2. Given any positive integer \(m\) exceeding \(1\), there are a prime number \(p\) such that \(p\equiv 1 \pmod{m}\) and \(d\) integers \(x_0,\dots ,x_{d-1}\) such that every element of the sequence \((x_n)\pmod{p}\) is not a square \(\pmod{p}\).

MSC:

11B37 Recurrences
11B50 Sequences (mod \(m\))
11T06 Polynomials over finite fields
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