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On the prime divisors of the sequence \(w_{n+1}=1+w_ 1\dots w_ n\). (English) Zbl 0574.10020
Consider the recursive sequence \(w_{n+1}=1+w_ 1...w_ n\), \(n\geq 1\), \(w_ 1=2\). This sequence, as encountered in Euclid’s proof of the infinity of the set of primes, can be rewritten as \(w_{n+1}=f(w_ n)\), \(f(X)=X^ 2-X+1\). The author is interested in the set P of primes dividing at least one of the \(w_ n\). By a very interesting application of Chebotarev’s density theorem the author shows, \[ P\cap [1,x]=O(x(\log x)^{-1}(\log \log \log x)^{-1})\quad as\quad x\to \infty. \]
Reviewer: F.Beukers

MSC:
11B37 Recurrences
11R45 Density theorems
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