A factorial prime is a prime number that is one more or one less than a factorial, i.e., it has the form:
p = n! ± 1
Where n! (n factorial) is the product of all the positive integers up to that number n, and p is a prime number.
First few factorial primes are:
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999, . . .
n! - 1 yields prime for following values:
n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003, . . .
n! + 1 yields prime for following values:
n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429, . . .
NOTE: For a number to be considered a factorial prime, it must meet the following criteria:
- It is derived from the n! ± 1.
- It is a prime number (only divisible by 1 and itself).
Properties of Factorial Primes
Some of the common properties of factorial primes are:
- All factorial primes derived from n! − 1 for n ≥ 3 are odd.
- The only small factorial primes are 5, 23, and 719 from the n!−1 form.
- They are directly related to factorial numbers, which are calculated as n!.
- Factorial primes are always derived from positive integers n.
- All factorials greater than 1 are even.
Conclusion
Factorial primes are a special type of prime number that come from adding or subtracting 1 from a factorial. Although they are rare, they are fascinating because they combine two important mathematical concepts: factorials and primes.
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