A079853 Primes p for which (p-2)! == 1 (mod p^2).

2, 3, 11, 107, 4931

Offset: 1

Author

Pavlos Saridis (pavlos19(AT)yahoo.com), Sep 13 2003

Keywords

• nonn
• more

Comments

These are generalized Wilson primes of order 2. Similarly to Wilson's theorem which states that (p-1)! == -1 (mod p) for every prime p>=n, we can prove that (n-1)!(p-n)! == (-1)^n (mod p) for every prime p. Generalized Wilson primes p of order n satisfy the recurrence (n-1)!(p-n)! == (-1)^n (mod p^2). Cf. A128666

Also, near-Wilson primes with Wilson quotient modulo p equals 1: prime p=prime(n) is in this sequence iff A002068(n) == A007619(n) == 1 (mod p).

Zhi-Wei SUN conjectures that for n>1, a(n) == 3 (mod 8). (Posting to the Number Theory Mailing List, Nov 02 2009; added by N. J. A. Sloane, Nov 02 2009)

No other terms below 4*10^11.

Conjecture: primes p such that Sum_{k=1..p-1} k^(1-p) == -1 (mod p^2) are the odd terms of this sequence. - Thomas Ordowski, Jul 02 2020

Links

• Wikipedia, Near-Wilson primes

Crossrefs

Cf. A002068, A007619, A128666.

Programs

Mathematica

Select[Prime[Range[700]],Mod[(#-2)!,#^2]==1&] (* Harvey P. Dale, Jun 01 2014 *)

PARI

forprime(n=2, 10^9, if(Mod((n-2)!, n^2)==1, print1(n, ", "))) \\ Felix Fröhlich, Jun 17 2014

Extensions

Edited by Max Alekseyev, Jan 28 2012