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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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