Factorial prime

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.
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Definition

A factorial prime is a prime that differs from a factorial by $1$. In other words, it is a prime of the form $n!\pm 1$.

Occurrence

Initial values

The initial values of factorial primes are given as:

2, 3, 5, 7, 23, 719, 5039, [SHOW MORE]

39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

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The first four primes $2,3,5,7$ are factorial primes. However, factorial primes become much rarer after that. The next two factorial primes are $23$ and $719$.

The initial values of $n$ for which $n!+1$ is prime are $n=1,2,3$. Note that, by Wilson's theorem, $n!+1$ cannot be prime if $n+1$ is prime, for $n\geq 3$. This explains, for instance, why $4!+1$ and $6!+1$ are not prime. $n=5,7,11$ are also Brown numbers -- they are solutions to Brocard's problem of $n!+1$ being a perfect square.

The initial values of $n$ for which $n!-1$ is prime are: $n=3,4,6,7$.