A197634 -id:A197634 - cp's OEIS frontend

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

Jonathan Sondow, Oct 19 2011

nonn

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

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

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.

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

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

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

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

A197633 Fermat-Wilson quotients of non-Wilson primes: q_p(w_p), where q_p(k) = (k^(p-1)-1)/p is a Fermat quotient, w_p = ((p-1)!+1)/p is a Wilson quotient, and p is a non-Wilson prime.

Original entry on oeis.org

0, 0, 170578899504, 1387752405580695978098914368989316131852701063520729400

Offset: 1

Jonathan Sondow, Oct 16 2011

nonn

A Wilson prime is a prime p that divides its Wilson quotient w_p (see A007619). The known Wilson primes are 5, 13, 563 (see A007540).
If p is a non-Wilson prime (see A197636), then p does not divide w_p, and so by Fermat's little theorem the Fermat quotient q_p(w_p) is an integer.
The next term is the Fermat-Wilson quotient of 17, which has 193 digits.
The Fermat-Wilson quotient of 14771 (see A197635) has over 800 million digits.
The GCD of all Fermat-Wilson quotients is 24. In particular, q_p(w_p) is never prime.

			The 3rd non-Wilson prime is 7, so a(3) = (((6!+1)/7)^6-1)/7 = 170578899504.

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
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.

Cf. A007540, A007619, A197634, A197635, A197636, A216867.

nmax=4; nonWilsonQ[p_] := Mod[((p-1)!+1)/p ,p] != 0; A197636 = Select[ Prime[ Range[nmax+2]], nonWilsonQ]; a[n_] := With[{p=A197636[[n]]}, ((((p-1)!+1)/p)^(p-1)-1)/p]; Table[ a[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 14 2011 *)

a(n) = ((((p-1)!+1)/p)^(p-1)-1)/p, where p = A197636(n).

A197635 Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.

Original entry on oeis.org

2, 3, 14771

Offset: 1

Jonathan Sondow, Oct 16 2011

A Wieferich prime base a is a prime p satisfying a^(p-1) == 1 (mod p^2). A non-Wilson prime p is called a Wieferich-non-Wilson prime if p is a Wieferich prime base w_p, where w_p = ((p-1)!+1)/p is the Wilson quotient of p.

Michael Mossinghoff has computed that if a 4th Wieferich-non-Wilson prime exists, it is > 10^7.

			The first two non-Wilson primes are 2 and 3, whose Fermat-Wilson quotients are 0. Since 2 and 3 divide 0, they are members.
The 1728th non-Wilson prime is prime(1731) = 14771, and A197634(1728) = 0, so 14771 is a member.

Carlos Rivera, Problem 59. Wieferich-non-Wilson primes, The Prime Puzzles and Problems Connection.
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
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.

Cf. A007540, A197633, A197634, A197636, A197637.

A197634(A197637(a(n))) = 0.

(((p-1)!+1)/p)^(p-1) == 1 (mod p^2), where p = a(n).

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

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A197633 Fermat-Wilson quotients of non-Wilson primes: q_p(w_p), where q_p(k) = (k^(p-1)-1)/p is a Fermat quotient, w_p = ((p-1)!+1)/p is a Wilson quotient, and p is a non-Wilson prime.

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A197635 Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.

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