# is it true that there are infinitely many primes $P$ of the form $p_0^{\alpha_0}p_1^{\alpha_1}p_2^{\alpha_2}…p_k^{\alpha_k}+1$?

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)

• 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. – ajotatxe Dec 14 '18 at 13:48
• "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. – Did Dec 14 '18 at 14:09
• @Did Sorry for my mistakes. I have deleted my comment. – apple Dec 14 '18 at 14:16