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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Apart from 2 and 5, subsequence of A045468.

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

Programs

Magma

[p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 5}]; // Vincenzo Librandi, Sep 11 2012

Mathematica

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