Factorial prime
Encyclopedia
A factorial prime is a prime number
that is one less or one more than a factorial
(all factorials above 1 are even). The first few factorial primes are:
n! − 1 is prime for :
n! + 1 is prime for :
No other factorial primes are known as of 13 June 2011.
Absence of primes to both sides of a factorial n! implies a relatively lengthy run of consecutive composite number
s, since n! ± k is divisible by k for 2 ≤ k ≤ n. For example, the next prime following 6227020777 = 13! − 23 is 6227020867 = 13! + 67 (a run of 89 consecutive composites); here the run is substantially longer than implied merely by the absence of factorial primes. Note that this is not the most efficient way to find large prime gaps. E.g., there are 95 consecutive composites between the primes 360653 and 360749.
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
that is one less or one more than a factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
(all factorials above 1 are even). The first few factorial primes are:
- 2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! − 1), 7 (3! + 1), 2323 (number)23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...
(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, ...
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, ...
No other factorial primes are known as of 13 June 2011.
Absence of primes to both sides of a factorial n! implies a relatively lengthy run of consecutive composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
s, since n! ± k is divisible by k for 2 ≤ k ≤ n. For example, the next prime following 6227020777 = 13! − 23 is 6227020867 = 13! + 67 (a run of 89 consecutive composites); here the run is substantially longer than implied merely by the absence of factorial primes. Note that this is not the most efficient way to find large prime gaps. E.g., there are 95 consecutive composites between the primes 360653 and 360749.
External links
- List of largest known factorial primes from the Prime PagesPrime pagesThe Prime Pages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms...
- Factorial Prime Search from PrimeGridPrimeGridPrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...