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It is known that for any two positive coprime integers $a$ and $d$, there are infinitely many primes of the form $a + nd$, where $n$ is a non-negative integer. This is known as the Dirichlet's theorem on arithmetic progressions.

So I am wondering, given $k+1$ different prime numbers: $p_0,p_1,...,p_k$, with $p_0=2$ and $k>0$, is it true that there are infinitely many primes $P$ such that all prime divisors of $P-1$ are only $p_0,p_1,...,p_k$ ? In other words, is it true that there are infinitely many primes $P$ of the form $p_0^{\alpha_0}p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}+1$ ?

(Please let me know if this question is off-topic or should be closed)

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    $\begingroup$ It is probably not known. For $k=0$ the question becomes "Are there infinitely many primes of the form $2^\alpha+1$?", which is an open problem. See Fermat's numbers, or Fermat's primes. $\endgroup$ – ajotatxe Dec 14 '18 at 13:48
  • $\begingroup$ "what about k>0 ?" You might have missed the intent of @ajotatxe's comment: what they are saying is that for no $k$ is this known, probably, and that already for $k=0$ it is not known. $\endgroup$ – Did Dec 14 '18 at 14:09
  • $\begingroup$ @Did Sorry for my mistakes. I have deleted my comment. $\endgroup$ – apple Dec 14 '18 at 14:16

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