A141305 Primes p such that q=(p-1)/2 is also prime and 2 is a primitive root mod q; that is, q is in A001122.

7, 11, 23, 59, 107, 167, 263, 347, 359, 587, 839, 887, 983, 1019, 1307, 1319, 2039, 2459, 2903, 2999, 3467, 3803, 3863, 3947, 4139, 4283, 4679, 4919, 5099, 5387, 5399, 5483, 5639, 5879, 5927, 6599, 6827, 6983, 7079, 7559, 7607, 7703, 8039, 8699, 8747

Offset: 1

Author

T. D. Noe, Jun 24 2008

Keywords

• nonn

Comments

These primes are a subset of the safe primes, A005385. These primes produce the longest possible cycles, (p-3)/2, in the squaring mod p map. See A037178.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Caroline Lucheta, Eli Miller and Clifford Reiter, Digraphs from Powers Modulo p, Fibonacci Quarterly, Volume 34, Number 3, June-July 1996. See p. 9.
• Troy Vasiga and Jeffrey Shallit, On the iteration of certain quadratic maps over GF(p), Discrete Mathematics, Volume 277, Issues 1-3, 28 February 2004, Pages 219-240. See p.9.

Crossrefs

Cf. A001122, A005385, A037178.

Programs

Mathematica

Select[Range[10^4], PrimeQ[#] && PrimeQ[(q = (# - 1)/2)] && PrimitiveRoot[q] == 2 &] (* Amiram Eldar, Oct 09 2021 *)

PARI

isok(p) = isprime(p) && (p%2) && isprime(q=(p-1)/2) && (q%2) && (znorder(Mod(2, q))==(q-1)); \\ Michel Marcus, Jan 30 2016

Extensions

Incorrect term 5 removed by Michel Marcus, Jan 30 2016