About the Primality of Primorials
George Lillie

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 where denotes the product of all primes less than or equal to , the -th prime. We show that the probability along the lines of Mertens' Theorem that either or is prime is and that the probability that both and are prime is , for . The latter result provides evidence that there are in total three instances where both and are prime. We provide proof that numbers of the from have the highest probability of being prime.
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Taxonomy
TopicsAnalytic Number Theory Research · Mathematics and Applications · History and Theory of Mathematics
