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A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number of prime Euclid numbers? The references on the papers in which people have tried partially answer the question are also accepted.

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I don't have the answer. But you may have a look to a related question of mine mathoverflow.net/questions/38794 –  Denis Serre Nov 11 '10 at 21:03
    
@Denis Serre Thank you, for the link. –  Oleksandr Bondarenko Nov 11 '10 at 21:14
    
You should expect more primes taking your number to be $ 3 \pmod 4$ and $ -1 \pmod q$ whenever you have prime $ q \equiv 3 \pmod 4,$ finally equivalent to your favorite quadratic nonresidue $\pmod p$ for each prime $p \equiv 1 \pmod 4.$ This is by the Chinese Remainder Theorem. Of course there is no attractive formula any more. Some articles have been published on this with part of the title being "primes races," in particular by Peter Sarnak, who may or may not have included the phrase. –  Will Jagy Nov 11 '10 at 21:18
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Its not a survey article, but your question is discussed in an article of Caldwell and Gallot: ams.org/journals/mcom/2002-71-237/S0025-5718-01-01315-1/… You might try looking at some of their references. –  Ben Linowitz Nov 11 '10 at 21:20
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According to Ribenboim (The little book of big primes, page 3), it is also not known whether there exist infinitely many composite Euclid numbers. Such questions are often very difficult to tackle. –  François Brunault Nov 12 '10 at 11:52
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