A045317 - OEIS

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

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OFFSET

1,1

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

MATHEMATICA

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

PROG

(Magma) [p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^8 eq 3}]; // 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

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.

Sequence in context: A164624 A215819 A040101 * A215378 A078763 A157884

Adjacent sequences: A045314 A045315 A045316 * A045318 A045319 A045320

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved