How often is $1+\prod_{k=1}^n p_k$ not prime?

Question

How often is $1+\prod_{k=1}^n p_k$ is not prime?,wkere $p_k$ is k'th prime consider that $2\times3\times5\times7\times11\times13+1=59\times509 = 30031$ is this one off or, are there infinitely many composites of the form $1+\prod_{k=1}^n p_k$?

Answer 1

Define $p_n\#=p_1\cdots p_n$. This is called the primorial of $p_n$ (like a factorial, but multiplying together only primes).

Prime numbers of the form $p_n\# \pm 1$ are called primorial primes. Along a similar vein, Euclid Numbers are of the form $p_n\#+1$ (not necessarily prime).

Thus, your question is: "How many prime Euclid Numbers are there?" The Wolfram Mathworld article says:

The largest known Euclid number is $E_{13494}$, and it is not known if there are an infinite number of prime Euclid numbers (Guy 1994, Ribenboim 1996).

Regarding composite Euclid numbers, this comment by François Brunault on MathOverflow says:

According to Ribenboim (The little book of big primes, page 3), it is also not known whether there exist infinitely many composite Euclid numbers. Such questions are often very difficult to tackle.

Summary: There are many known prime Euclid Numbers (A014545 is a list of $n$ such that $E_n$ is prime), but it is not known if there are infinitely many such primes. There are also many known composite Euclid Numbers, but it is not known if there are infinitely many.