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Is there an infinite number of primes constructed as in Euclid's proof?

The question is :

Are there infinitely many primes of the form $p_1\cdot p_2\cdot...p_n+1$? ($p_k$ is the $k$-th prime.)

For example: $2\cdot3 + 1$.

But $2\cdot5+1$ is not included in the set of primes that i want demostrate, because 2 and 5 are not primes consecutive ... sorry for my English and thanks in advance


marked as duplicate by Beni Bogosel, J. M. is a poor mathematician, joriki, Asaf Karagila, Srivatsan Nov 23 '11 at 9:19

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    $\begingroup$ a number of the form $p_1\cdot p_2 \cdot... \cdot p_n$ cannot be prime. Maybe you meant $p_1\cdot p_2 \cdot... \cdot p_n+1$ $\endgroup$ – Beni Bogosel Nov 23 '11 at 8:58
  • $\begingroup$ @Beni: Mauricio did slip up in the body, but the error in the title is not his fault. I'll fix. $\endgroup$ – J. M. is a poor mathematician Nov 23 '11 at 9:01
  • $\begingroup$ yes Beni Bogosel i mean p1*p2*...*pn+1 $\endgroup$ – titusfx Nov 23 '11 at 9:18

Ribenboim and Guy claim that it is not yet known whether there are infinitely many primorial primes, $\left(\prod\limits_{k=1}^n p_k\right)\pm 1$.


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