

A277167


Prime numbers p such that (1)^h + (h!)^2 == 0 (mod p^2) where h = (p1)/2.


0




OFFSET

1,1


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


REFERENCES

Lagrange, "Démonstration d’un théoreme nouveau concernant les nombres premiers," Nouveaux Mémoires de l’Académie Royale des Sciences et BellesLettres [de Berlin], année 1771 (published 1783), 125137.
G. B. Mathews, Theory of Numbers, part 1 [all published] (Cambridge, 1892), 318.


LINKS



EXAMPLE

(1)^((111)/2)+(((111)/2)!)^2 = 14399 = 7*11^2*17.


PROG

(PARI) lista(nn) = forprime(p=3, nn, if ((((1)^((p1)/2)+(((p1)/2)!)^2) % p^2) == 0, print1(p, ", "))); \\ Michel Marcus, Oct 02 2016


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



