Factorial prime
From Number
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, 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: 
.
