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A {\em 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: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 (sequence A088054 in the OEIS). It is conjectured that only for $n = 3$ are both $n! - 1$ and $n! + 1$ both primes.
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A {\em 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: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 (sequence A088054 in the OEIS)..
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| Factorial primes have a r\^ole in an argument that 1 is not a prime number. If $n$ is a positive integer 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 if we subtract instead of add). |
Factorial primes have a r\^ole in an argument that 1 is not a prime number. If $n$ is a positive integer 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 if we subtract instead of add). |
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