A045317 Primes p such that x^8 = 3 has a solution mod p.

2, 3, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of A045318 relative to A000040. - Vincenzo Librandi, Sep 13 2012

Union of 2, 5, A068231 (primes congruent to 11 modulo 12), prime p == 5 (mod 8) such that 3^((p-1)/4) == 1 (mod p), and primes p == 1 (mod 8) such that 3^((p-1)/8) == 1 (mod p). - Jianing Song, Jun 22 2025

Links

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

Crossrefs

Cf. A000040, A045318.

A068231 < A385220 < this sequence < A040101 < 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^8 eq 3}]; // Vincenzo Librandi, Sep 13 2012

Mathematica

ok[p_]:= Reduce[Mod[x^8- 3, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

PARI

isok(p) = isprime(p) && ispower(Mod(3, p), 8); \\ Michel Marcus, Oct 17 2018

PARI

isA045317(p) = isprime(p) && (p==2 || p==3 || p%12==11 || (p%8==5 && Mod(3, p)^((p-1)/4) == 1) || (p%8==1 && Mod(3, p)^((p-1)/8) == 1)) \\ Jianing Song, Jun 22 2025