A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

2, 3, 7, 11, 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, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 569

Offset: 1

Author

Jonathan Sondow, Oct 19 2011

Keywords

• nonn

Comments

All primes except 5, 13, 563, and any other Wilson prime A007540 that may exist.

Same as primes p that do not divide their Wilson quotient ((p-1)!+1)/p.

Wilson's theorem says that (p-1)! == -1 (mod p) if and only if p is prime.

p = prime(i) is a term iff A250406(i) != 0. - Felix Fröhlich, Jan 24 2016

Complement of A007540 in A000040. - Felix Fröhlich, Jan 24 2016

Examples

			2 is a non-Wilson prime since (2-1)! = 1 ==/== -1 (mod 2^2).
		

Links

• E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:1209.3436 [math.NT], 2012).
• Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
• Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Crossrefs

Cf. A007540, A007619, A197633, A197634, A197635, A250406.

Programs

Mathematica

Select[Prime@ Range@ 104, Mod[Factorial[# - 1], #^2] != #^2 - 1 &] (* Michael De Vlieger, Jan 24 2016 *)

PARI

forprime(p=1, 500, if(Mod((p-1)!, p^2)!=-1, print1(p, ", "))) \\ Felix Fröhlich, Jan 24 2016

Formula

((p-1)!+1)/p =/= 0 (mod p), where p is prime.