A058302 Primes p such that p | ((p-1)/2)! -1.

3, 23, 31, 59, 71, 83, 107, 139, 151, 167, 211, 223, 239, 251, 271, 283, 307, 311, 331, 359, 379, 439, 463, 467, 487, 499, 547, 587, 643, 647, 659, 719, 751, 811, 827, 859, 883, 907, 911, 919, 967, 971, 983, 1031, 1039, 1063, 1103, 1163, 1171, 1223

Offset: 1

Author

Robert G. Wilson v, Dec 08 2000

Keywords

• nonn

Comments

p | (p-1)! +1 iff p is a prime (Wilson's theorem). All of the above primes are congruent to 3 (mod 4).

Primes p such that p | ((p-3)/2)! +2. - Davide Rotondo, Jun 03 2024

References

• J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.5.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
• Math Overflow, Primes P such that ((P-1)/2)!=1 mod P [From _T. D. Noe_, Feb 24 2010]

Crossrefs

Cf. A002145, A055939.

Programs

Magma

[p: p in PrimesInInterval(3,1230) | IsDivisibleBy(Factorial((p-1) div 2)-1, p)];  // Bruno Berselli, Apr 13 2011

Mathematica

Select[ Range[ 1225 ], PrimeQ[ # ] && Mod[ ((# - 1)/2)! - 1, # ] == 0 & ]
Select[Prime[Range[200]],Divisible[((#-1)/2)!-1,#]&] (* Harvey P. Dale, Aug 29 2022 *)

PARI

forprime(p=3,10^4,if( Mod(((p-1)/2)!,p)==1,print1(p,", "))); /* Joerg Arndt, Apr 12 2011 */

PARI

is(p)=isprime(p) && p%4==3 && if(p>9, qfbclassno(-p)%4, p)==3 \\ Charles R Greathouse IV, Nov 04 2013