A040101 Primes p such that x^4 = 3 has a solution mod p.

2, 3, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 193, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601, 647, 659

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Union of 2, 3, A068231 (primes congruent to 11 modulo 12), primes p == 1 (mod 4) such that 3^((p-1)/4) == 1 (mod p). - Jianing Song, Jun 22 2025

Links

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

Crossrefs

A subsequence of A038874.

A068231 < A385220 < A045317 < this sequence < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.

Programs

Magma

[p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 3}]; // Vincenzo Librandi, Sep 11 2012

Mathematica

ok [p_]:=Reduce[Mod[x^4- 3, p] == 0, x, Integers] =!= False;  Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 11 2012 *)

PARI

isA040101(p) = isprime(p) && (p==2 || p==3 || p%12==11 || (p%4==1 && Mod(3, p)^((p-1)/4) == 1)) \\ Jianing Song, Jun 22 2025