A320384 - OEIS

%I #14 Oct 13 2018 15:36:26

%S 7,11,17,31,37,41,59,83,89,103,107,109,113,127,131,137,151,157,179,

%T 223,227,229,233,251,257,271,277,347,349,353,367,397,421,443,449,467,

%U 491,521,541,563,569,587,593,607,613,631,641,659,661,683,733,757,761,809,827,853,857,877,929,953,967,971,977,991

%N Primes p such that 3/2 is a primitive root modulo p.

%C Primes p such that the smallest positive k such that p divides 3^k - 2^k is p - 1.

%C All terms are congruent to 7, 11, 13, 17 modulo 24. For other primes p, 3/2 is a quadratic residue modulo p.

%C By Artin's conjecture, this sequence contains 37.395% of all primes, or 74.79% of all primes congruent to 7, 11, 13, 17 modulo 24.

%H C. Hooley, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002182270">On Artin's conjecture</a>, J. reine angewandte Math., 225 (1967) 209-220.

%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>

%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Primitive_root_modulo_n">Primitive root modulo n</a>

%H <a href="/index/Ar#Artin">Index entries for sequences related to Artin's conjecture</a>

%H <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>

%e 3/2 == 5 (mod 7), 5 is a primitive root modulo 7, so 7 is a term. Indeed, 7 does not divide 3^2 - 2^2 or 3^3 - 2^3, but it divides 3^6 - 2^6.

%e 3/2 == 7 (mod 11), 7 is a primitive root modulo 11, so 11 is a term. Indeed, 11 does not divide 3^2 - 2^2 or 3^5 - 2^5, but it divides 3^10 - 2^10.

%e 3/2 == 13 (mod 23), 13^11 == 1 (mod 23), so 23 is not a term. Indeed, 23 divides 3^11 - 2^11.

%o (PARI) forprime(p=5,10^3,if(p-1==znorder(Mod(3/2,p)),print1(p,", "))); \\ _Joerg Arndt_, Oct 13 2018

%Y Cf. A019336, A320383.

%K nonn

%O 1,1

%A _Jianing Song_, Oct 12 2018