Factorial prime
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A factorial prime is a number that is one less or one more than a factorial and is also a prime number. The first few factorial primes are:
Factorial primes are of interest to number theorists because their absence sometimes signals the end or the beginning of a lengthy run of consecutive composite numbers. For example, the prime following 6227020777 is 6227020867.
Factorial primes have a role in the argument that 1 is not a prime number.
If n is a natural number and p is a prime number, n! + p is never a prime for p < n, because obviously it will be a multiple of p, just as n! is. But n! + 1, even though it certainly is a multiple of 1, can be a prime, specifically, a factorial prime. (The same is also true for n! - p and n! - 1).
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