A323705 Odd primes p such that ((p+1)/2)^(p-1) == 1 (mod p^2).

1897121, 52368101, 126233057

Offset

1,1

Comments

Primes p such that the Fermat quotient of p base 2 (A007663) is congruent to -1 modulo p. In other words, 2^(p-1) == 1-p (mod p^2). - Max Alekseyev, May 09 2026

The corresponding sequence when (p+1) in the congruence is replaced with (p-1) is A125854.

a(4) > 4262937421 if it exists.

Links

Table of n, a(n) for n=1..3.

Mathematica

Select[Prime[Range[718*10^4]], PowerMod[(#+1)/2, #-1, #^2]==1&] (* Harvey P. Dale, May 14 2021 *)

Prog

(PARI) forprime(p=3, , my(x=(p+1)/2); if(Mod(x, p^2)^(p-1)==1, print1(p, ", ")))

Crossrefs

Cf. A007663, A125854, A259502, A355959.

Sequence in context: A032597 A203794 A204672 * A249318 A194620 A182360

Adjacent sequences: A323702 A323703 A323704 * A323706 A323707 A323708

Keyword

nonn,hard,bref,more

Author

Felix Fröhlich, Jan 24 2019

Status

approved