User:Peter Luschny/SwingingPrimes

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)^⌊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. 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).		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:
Literature
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., 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.  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.