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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