Factorial prime

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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 . In other words, it is a prime of the form .

Occurrence

Initial values

The initial values of factorial primes are given as: 2, 3, 5, 7, 23, 719, 5039, [SHOW MORE] View list on OEIS

The first four primes are factorial primes. However, factorial primes become much rarer after that. The next two factorial primes are and .

The initial values of for which is prime are . Note that, by Wilson's theorem, cannot be prime if is prime, for . This explains, for instance, why and are not prime. are also Brown numbers -- they are solutions to Brocard's problem of being a perfect square.

The initial values of for which is prime are: .