Swinging Primes Factorial primes, primes which are within 1 of a factorial number (n! A000142). Swinging primes, primes which are within 1 of a swinging factorial number (n≀ A056040). On this page `?´ is a meta symbol denoting either `!´ or `≀´. !  A088054 ≀  A163074 A74 := proc(f,n) select(isprime, map(x -> f(x)+1,[\$1..n])); select(isprime, map(x -> f(x)-1,[\$1..n])); sort(convert(convert(%%,set) union convert(%,set),list)) end: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199 2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011, 48619, 51479, 51481, 2704157 Primes of the form n? + 1 !  A088332 ≀  A163075 A75 := proc(f,n) select(isprime, map(x -> f(x)+1,[\$1..n])) end: 2, 3, 7, 39916801, 10888869450418352160768000001 2, 3, 7, 31, 71, 631, 3433, 51481, 2704157 Primes of the form n? - 1. !  A055490 ≀  A163076 A76 := proc(f,n) select(isprime, map(x -> f(x)-1,[\$1..n])); sort(%) end: 5, 23, 719, 5039, 479001599, 87178291199 5, 19, 29, 139, 251, 12011, 48619, 51479, 155117519 Numbers n such that n? + 1 is prime. !  A002981 ≀  A163077 A77 := proc(f,n) select(x -> isprime(f(x)+1),[\$0..n]) end: 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320 0, 1, 2, 3, 4, 5, 8, 9, 14, 15, 24, 27, 31, 38, 44 Numbers n such that n? - 1 is prime. !  A002982 ≀  A163078 A78 := proc(f,n) select(x -> isprime(f(x)-1),[\$0..n]) end: 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324 3, 4, 5, 6, 7, 10, 13, 15, 18, 30, 35, 39, 41, 47 Primes p such that p? + 1 is also prime. !  A093804 ≀  A163079 A79 := proc(f,n) select(isprime, select(k -> isprime(f(k)+1),[\$0..n])) end: 2, 3, 11, 37, 41, 73, 26951 2, 3, 5, 31, 67, 139, 631 Primes p such that p? - 1 is also prime. !  A103317 ≀  A163080 A80 := proc(f,n) select(isprime, select(k -> isprime(f(k)-1),[\$0..n])) end: 3, 7, 379, 6917 3, 5, 7, 13, 41, 47, 83, 137, 151, 229, 317, 389, 1063 Primes of the form p? + 1 where p is prime. !  A103319 ≀  A163081 A81 := proc(f,n) select(isprime,[\$2..n]); select(isprime, map(x -> f(x)+1,%)) end: 3, 7, 39916801 3, 7, 31, 4808643121, 483701705079089804581 Primes of the form p? - 1 where p is prime. ! A000000 ≀  A163082 A82 := proc(f,n) select(isprime,[\$2..n]); select(isprime, map(x -> f(x)-1,%)) end: 5, 5039 5, 29, 139, 12011, 5651707681619, 386971244197199 Primes of the form p? + 1 which are the greater of twin primes. !  A000000 ≀  A163083 A83 := proc(f,n) select(s->isprime(s) and isprime(s-2), map(k->f(k)+1,[\$4..n])) end; 7 7, 31, 51481, 1580132580471901
 Al-Haytham Primes Al-Haytham is the first person that we know to state:  If p is prime then (p−1)! + 1 is divisible by p. Origin unknown to the author: If p is prime then (p−1)≀ − (−1)^floor(p/2) is divisible by p. Wilson quotients: ((p − 1)? + r(p)) / p, p prime !  A007619 ≀  A163210 WQ := proc(f,r,n) map(p->(f(p-1)+r(p))/p, select(isprime,[\$1..n])) end: WQ(factorial,p->1,30); WQ(swing,p->(-1)^iquo(p+2,2),30); 1, 1, 5, 103, 329891, 36846277, 1230752346353 1,1,1,3,23,71,757,2559,30671,1383331, 5003791 Wilson quotients which are prime !  A163212 ≀  A163211 WQP := proc(f,r,n) select(isprime,WQ(f,r,n)) end: WQP(factorial,p->1, 30); WQP(swing,p->(-1)^iquo(p+2,2), 40); 5, 103, 329891, 10513391193507374500051862069 3, 23, 71, 757, 30671, 1383331, 245273927 Wilson remainders: ((p − 1)? + r(p)) / p mod p, p prime !  A002068 ≀  A163213 WR := proc(f,r,n) map(p->(f(p-1)+r(p))/p mod p, select(isprime,[\$1..n])) end: WR(factorial,p->1, 36); WR(swing,p->(-1)^iquo(p+2,2), 36); 1, 1, 0, 5, 1, 0, 5, 2, 8, 18, 19, 7, 16, 13 1, 1, 1, 3, 1, 6, 9, 13, 12, 2, 19 Wilson primes: (((p − 1)? + r(p)) / p) mod p = 0 !  A007540 ≀  A001220 WP := proc(f,r,n) select(p->(f(p-1)+r(p))/p mod p = 0, select(isprime,[\$1..n])) end: WP(factorial,p->1, 600); WP(swing,p->(-1)^iquo(p+2,2), 3600); 5, 13, 563 1093, 3511 Wilson spoilers: composite n which divide (n − 1)? + r(n) !  A00000 ≀  A163209 WS := proc(f,r,n) select(p->(f(p-1)+r(p))mod p = 0,[\$2..n]); select(q -> not isprime(q),%) end: WS(factorial,p->1, 600); WS(swing,p->(-1)^iquo(p+2,2), 6000); There are none, as proved by Lagrange. 5907, 1194649, 12327121 Notation Replace '?' by '!' in the formulas and 'f' by 'factorial' in the Maple call proc(f, n) if you want to compute primes related to the factorial function, or replace '?' by '≀' in the formulas and 'f' by 'swing' in the Maple call if you want to refer to the swinging factorial function. Here 'swing' is the function in the box at the right hand side (see A056040). A Maple worksheet is here. Swinging factorials and swinging primes have been studied in: Peter Luschny, Divide, swing and conquer the factorial and the lcm{1,2,...,n}, preprint, April 2008. swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:

T. Agoh, On Bernoulli and Euler numbers, Manuscripta Math. 61 (1988), 1-10.
T. Agoh, K. Dilcher, and L. Skula, Fermat quotients for composite moduli, J. Number Theory 66 (1997), 29-50.
T. Agoh, K. Dilcher, and L. Skula, Wilson Quotients for Composite Moduli, Math. Comp. 67 (1998), 843-861.
R. E. Crandall, Topics in Advanced Scientific Computation, TELOS/Springer-Verlag, Santa Clara, CA, 1996.
R. E. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), 433-449.
L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea Pub. Company, N.Y., 1966.
H. Dubner, Searching for Wilson primes, J. Recreational Math. 21 (1989), 19-20.
R. H. Gonter and E. G. Kundert, All prime numbers up to 18,876,041 have been tested without finding a new Wilson prime, Preprint (1994).
K. E. Kloss, Some number theoretic calculations, J. Res. Nat. Bureau of Stand., B, 69 (1965), 335-339.
M. Lerch, Zur Theorie des Fermatschen Quotienten, Math. Annalen 60 (1905), 471-490.
E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350-360.
P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1988.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991.