About the Primality of Primorials

TL;DR

This paper investigates the primality of primorial numbers of the form p_n# ± 1, showing that such numbers have decreasing probabilities of being prime as n increases, and provides evidence for only three such twin prime instances.

Contribution

It establishes probabilistic bounds for primorial primes and proves that numbers of the form p_n# ± 1 are most likely to be prime among similar constructs.

Findings

Probability that p_n# ± 1 are prime decreases as n increases

Both p_n# - 1 and p_n# + 1 being prime is extremely rare, with at most three known instances

Numbers of the form p_n# ± 1 have the highest chance of being prime among similar forms

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.