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Let $p_n$ be the $n$-th prime, numbers $E_n=p_1\cdot\ldots\cdot p_n+1$ are called Euclid numbers.

https://en.wikipedia.org/wiki/Euclid_number

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?

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No. Check the notes here. The first example is $E_7$ and $E_{17}$...

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