Factorial prime
| No.of known terms | 52 |
|---|---|
| Conjecturedno.of terms | Infinite |
| Subsequenceof | n!± 1 |
| First terms | 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 |
| Largest known term | 422429! + 1 |
| OEISindex | A088054 |
Afactorial primeis aprime numberthat is one less or one more than afactorial(all factorials greater than 1 areeven).[1]
The first 10 factorial primes (forn= 1, 2, 3, 4, 6, 7, 11, 12, 14) are (sequenceA088054in theOEIS):
- 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 (sequenceA002982in theOEIS):
- 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 (sequenceA002981in theOEIS):
- 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 as of October 2022[update].
When bothn!+ 1 andn!− 1 arecomposite,there must be at least 2n+ 1 consecutive composite numbers aroundn!, since besidesn!± 1 andn!itself, also, each number of formn!±kisdivisiblebykfor 2 ≤k≤n.However, the necessary length of this gap is asymptotically smaller than the average composite run forintegersof similar size (seeprime gap).
See also
[edit]External links
[edit]- Weisstein, Eric W."Factorial Prime".MathWorld.
- The Top Twenty: Factorial primesfrom thePrime Pages
- Factorial Prime SearchfromPrimeGrid