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 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 (n-1) \times (n-2) \ ...
(all factorials greater than 1 are
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
* Odd and Even, a solitaire game wh ...
).
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, ... (resulting in 27 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
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials ...
, 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 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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s of similar size (see
prime gap).
See also
*
Primorial prime
External links
*
The Top Twenty: Factorial primesfrom the
Prime Pages
Factorial Prime Searchfrom
PrimeGrid
References
{{Prime number classes, state=collapsed
Integer sequences
Classes of prime numbers
Factorial and binomial topics