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Given $k+1$ different prime numbers: $p_0,p_1,...,p_k$, with $p_0=2$ and $k>0$, is it true that there exist at least a prime $P$ such that all prime divisors of $P-1$ are only $p_0,p_1,...,p_k$ ? In other words, is it true that there is a prime $P$ of the form $p_0^{\alpha_0}p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}+1$ ?

I know this should be an open problem if I asked there exist infinitely many primes $P$ (I have asked here: is it true that there are infinitely many primes $P$ of the form $p_0^{\alpha_0}p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}+1$? , I may delete this question), since it is unclear whether there are infinitely many primes of the form $2^\alpha +1$ or not (Fermat's prime). However, there is a least a prime of the form $2^\alpha +1$, is $5$.

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

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