A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

3, 11, 31, 47, 53

Offset: 1

Author

René Gy, Oct 01 2016

Keywords

• nonn
• hard
• more

Comments

The above congruence is true modulo p for all odd primes. See A089043. But like for Wilson congruence, it is true modulo p^2, for a restricted number of primes. After 53, the next one (if any) seems very far away (>500000).

The fact that the congruence is true modulo p for all odd primes was proved by Lagrange in 1771. Using a theorem of Mathews (1892) and Eisenstein's logarithmetic rule for the Fermat quotient, the condition stated in the definition can be restated as W_p == -2q_p(2) (mod p), where W_p is the Wilson quotient of p (A007619) and q_p(2) is the Fermat quotient of p, base 2 (A007663). - John Blythe Dobson, Jul 31 2017

Examples

			(-1)^((11-1)/2)+(((11-1)/2)!)^2 = 14399 = 7*11^2*17.
		

References

• Lagrange, "Démonstration d’un théoreme nouveau concernant les nombres premiers," Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres [de Berlin], année 1771 (published 1783), 125-137.
• G. B. Mathews, Theory of Numbers, part 1 [all published] (Cambridge, 1892), 318.

Links

• René Gy, Find the next "Wilson-like" prime.

Crossrefs

Cf. A007540, A089043.

Programs

PARI

lista(nn) = forprime(p=3, nn, if ((((-1)^((p-1)/2)+(((p-1)/2)!)^2) % p^2) == 0, print1(p, ", "))); \\ Michel Marcus, Oct 02 2016