A152217 Primes p == 1 (mod 3) such that ((p-1)/3)! == 1 (mod p).

3571, 4219, 13669, 25117, 55897, 89269, 102121, 170647, 231019, 246247, 251431

Offset

1,1

Comments

The Wilson theorem states that p is prime if and only if (p-1)! = -1 (mod p). If p = 3 (mod 4) then ((p-1)/2)! = +/- 1 (mod p).

References

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.2.

Links

Table of n, a(n) for n=1..11.

J. B. Cosgrave, Jacobi [From Francois Brunault (brunault(AT)gmail.com), Nov 29 2008]

Wikipedia, Wilson's theorem

Example

For n = 1 the prime a(1) = 3571 divides 1190! - 1.

Prog

(PARI) forprime(p=2, 30000, if(p%3==1 & ((p-1)/3)!%p==1, print(p)))

Crossrefs

Seems to be a subsequence of A002407 and therefore of A003215 (differences of consecutive cubes). See also A058302 and A055939 for the sequences corresponding to ((p-1)/2)! = +/- 1 (mod p).

Sequence in context: A204417 A204410 A204409 * A004932 A004952 A004972

Adjacent sequences: A152214 A152215 A152216 * A152218 A152219 A152220

Keyword

nonn,more

Author

Francois Brunault (brunault(AT)gmail.com), Nov 29 2008, Nov 30 2008

Status

approved