# is it true that there **exist at least** a prime $P$ of the form $p_0^{\alpha_0}p_1^{\alpha_1}p_2^{\alpha_2}…p_k^{\alpha_k}+1$ ?

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)