A040105 Primes p such that x^4 = 5 has a solution mod p.

2, 5, 11, 19, 31, 59, 71, 79, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 239, 251, 269, 271, 311, 331, 359, 379, 389, 401, 409, 419, 431, 439, 449, 461, 479, 491, 499, 521, 541, 569, 571, 599, 619

Offset

1,1

Comments

Union of 2, 5, A122869 (primes congruent to 11 or 19 modulo 20), and primes p == 1 (mod 4) such that 5^((p-1)/4) == 1 (mod p). - Jianing Song, Jun 20 2025

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Mathematica

ok [p_]:=Reduce[Mod[x^4- 5, p] == 0, x, Integers] =!= False;  Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 11 2012 *)

Prog

(Magma) [p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 5}]; // Vincenzo Librandi, Sep 11 2012
(PARI) isA040105(p) = isprime(p) && (p==2 || p==5 || p%20==11 || p%20==19 || (p%4==1 && Mod(5, p)^((p-1)/4) == 1)) \\ Jianing Song, Jun 20 2025

Crossrefs

Apart from 2 and 5, subsequence of A045468.

A385192 (which itself contains A122869) is a proper subsequence.

Sequence in context: A019341 A327265 A084572 * A319859 A156768 A134694

Adjacent sequences: A040102 A040103 A040104 * A040106 A040107 A040108

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved