For given finite prime numbers set $P$, does there exist some prime number $p$ such that for any $\ell\in P$, $\ell\mid (p+1)$?
For example, if $P=\{2,3,7\}$, then we can take $p=41$. In this case, $(\prod_{\ell\in P}\ell)-1$ is already prime number, so it's simple.
Even if this is not the case, e.g. $P=\{5,7\}$, we can take $p=f\cdot (\prod_{\ell\in P}\ell)-1$ where $f=4, p=139$.
I consider there exists counterexample, but giving the example is difficult since for
$p=826791736418446924644415105270960270928927659729776400179861442336062222833458285859,$
$(p+1)$ has all under $200$ primes as prime factor.
Especially, I want to know the case $P_{\le B}:=\{\ell~|~\ell \le B,~ \ell~ {\rm is ~prime }\}$ where $B$ is some integer.
Please help me.