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

1,1

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

Mathematica

ok [p_]:=Reduce[Mod[x^4- 3, 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 3}]; // 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

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.

Sequence in context: A038874 A164624 A215819 * A045317 A215378 A078763

Adjacent sequences: A040098 A040099 A040100 * A040102 A040103 A040104

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved