A019335 Primes with primitive root 5.

2, 3, 7, 17, 23, 37, 43, 47, 53, 73, 83, 97, 103, 107, 113, 137, 157, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 317, 347, 353, 373, 383, 397, 433, 443, 463, 467, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 617, 647, 653, 673, 677, 683, 727

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

To allow primes less than the specified primitive root m (here, 5) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019

Appears to be the numbers k such that the sequence 5^n mod k has period length k-1. All terms are congruent to 2 or 3 mod 5. - Gary Detlefs, May 21 2014

From Jianing Song, Apr 27 2019: (Start)

If we define

Pi(N,b) = # {p prime, p <= N, p == b (mod 5)};

Q(N) = # {p prime, p <= N, p in this sequence},

then by Artin's conjecture, Q(N) ~ (20/19)*C*N/log(N) ~ (40/19)*C*(Pi(N,2) + Pi(N,3)), where C = A005596 is Artin's constant.

Conjecture: if we further define

Q(N,b) = # {p prime, p <= N, p == b (mod 5), p in this sequence},

then we have:

Q(N,2) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,2);

Q(N,3) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,3). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Artin's constant
• Wikipedia, Artin's conjecture on primitive roots
• Index entries for primes by primitive root

Crossrefs

Cf. A019334-A019421.

Programs

Mathematica

pr=5; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

PARI

isok(p) = isprime(p) && (p != 5) && (znorder(Mod(5, p)) == p-1); \\ Michel Marcus, Apr 27 2019