$\displaystyle \prod_{i=1}^n p_i - 1$ is called Euclid number of the second kind (or Kummer number) , where $p_i$ is the i-th prime number.
It is not known whether there are infinitely many prime Kummer numbers. https://en.wikipedia.org/wiki/Euclid_number
Is it also unknown whether there are infinitely many composite Kummer numbers ?