About the Primality of Primorials

Abstract

A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either $p_n\# -1$ or $p_n\# +1$ is prime is $O(n^{-1})$ and that the probability that both $p_n\# -1$ and $p_n\# +1$ are prime is $O(n^{-2})$, for $n>1$. The latter result provides evidence that there are in total three instances where both $p_n\# -1$ and $p_n\# +1$ are prime. We provide proof that numbers of the from $p_n\# \pm 1$ have the highest probability of being prime.