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

1,1

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.

Links

Table of n, a(n) for n=1..64.

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

Example

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.

Prog

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

Crossrefs

Cf. A019336, A320383.

Sequence in context: A078725 A247478 A174360 * A019418 A068674 A156112

Adjacent sequences: A320381 A320382 A320383 * A320385 A320386 A320387

Keyword

nonn

Author

Jianing Song, Oct 12 2018

Status

approved