Factorial prime

Factorial prime
No. of known terms	52
Conjectured no. of terms	Infinite
Subsequence of	n! ± 1
First terms	2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199
Largest known term	422429! + 1
OEIS index	A088054

A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even).^[1]

The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (sequence A088054 in the OEIS):

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 (sequence A002982 in the OEIS):

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 (sequence A002981 in the OEIS):

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 both n! + 1 and n! − 1 are composite, there must be at least 2n + 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 integers of similar size (see prime gap).

YouTube Encyclopedic

1/3
Views:
2 145
422
816

Transcription

External links

Weisstein, Eric W. "Factorial Prime". MathWorld.
The Top Twenty: Factorial primes from the Prime Pages
Factorial Prime Search from PrimeGrid

References

^ "Weisstein, Eric W. "Factorial Prime." From MathWorld".

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

This page was last edited on 19 July 2023, at 09:49

From Wikipedia, the free encyclopedia

YouTube Encyclopedic

Transcription

See also

External links

References