A factorial prime is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
that is one less or one more than a
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
(all factorials greater than 1 are
even).
The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are :
:
2 (0! + 1 or 1! + 1),
3 (2! + 1),
5 (3! − 1),
7 (3! + 1),
23 (4! − 1), 719 (6! − 1), 5039 (7! − 1), 39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1), ...
''n''! − 1 is prime for :
:''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, 632760, ... (resulting in 28 factorial primes)
''n''! + 1 is prime for :
:''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, ... (resulting in 24 factorial primes - the prime 2 is repeated)
No other factorial primes are known .
When both ''n''! + 1 and ''n''! − 1 are
composite, there must be at least 2''n'' + 1 consecutive composite numbers around ''n''!, since besides ''n''! ± 1 and ''n''! itself, also, each number of form ''n''! ± ''k'' is
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
by ''k'' for 2 ≤ ''k'' ≤ ''n''. However, the necessary length of this gap is asymptotically smaller than the average composite run for
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s of similar size (see
prime gap).
See also
*
Primorial prime
External links
*
The Top Twenty: Factorial primesfrom the
Prime Pages
The PrimePages is a website about prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is ...
Factorial Prime Searchfrom
PrimeGrid
References
{{Prime number classes, state=collapsed
Integer sequences
Classes of prime numbers
Factorial and binomial topics