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On prime divisors of linear recurrent sequences. (Sur les diviseurs premiers des suites récurrentes linéaires.) (French) Zbl 0635.10004
Let $$u(n)=\sum_{k=1}^s a_kb^n_k$$, $$n\in\mathbb N$$ be a recurrent sequence of integers, where $$a_k$$, $$b_k$$ are non-zero rational integers and the $$b_k$$ are all distinct. Let $$\varepsilon(n)$$ be a positive function tending to zero as $$n$$. Let $$P(N)$$ denote the greatest prime factor of $$N\in\mathbb N$$. Then the author proves that $$P(u(n))\geq \varepsilon(n)n^{1/3}$$ for almost all $$n\in\mathbb N$$, that is, the number of exceptions to this inequality has arithmetic density zero. This very pretty theorem is proved by elementary means. Theorem 2 deals with the case $$s=2$$ and gives a lower bound for the asymptotics of the set of primes dividing the $$u(n)$$’s.
##### MSC:
 11B37 Recurrences
Full Text:
##### References:
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