A320384 Primes p such that 3/2 is a primitive root modulo p.

7, 11, 17, 31, 37, 41, 59, 83, 89, 103, 107, 109, 113, 127, 131, 137, 151, 157, 179, 223, 227, 229, 233, 251, 257, 271, 277, 347, 349, 353, 367, 397, 421, 443, 449, 467, 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

Offset: 1

Author

Jianing Song, Oct 12 2018

Keywords

• nonn

Comments

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

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

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.

Examples

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

Links

• C. Hooley, On Artin's conjecture, J. reine angewandte Math., 225 (1967) 209-220.
• Wikipedia, Artin's conjecture on primitive roots
• Wikipedia, Primitive root modulo n
• Index entries for sequences related to Artin's conjecture
• Index entries for primes by primitive root

Crossrefs

Cf. A019336, A320383.

Programs

PARI

forprime(p=5,10^3,if(p-1==znorder(Mod(3/2,p)),print1(p,", "))); \\ Joerg Arndt, Oct 13 2018