Let $p_n$ be the $n$-th prime, numbers $E_n=p_1\cdot\ldots\cdot p_n+1$ are called Euclid numbers.
It is not known if there are infinitely many primes among $E_n$'s.
The second open question is if all Euclid numbers are squarefree. Are Euclid numbers squarefree?
Are any two distinct Euclid numbers relatively prime? Is this known?