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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
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