A385188 Primes p == 3 (mod 4) such that the multiplicative order of 2+-i modulo p in Gaussian integers (A385165) is not divisible by 2 or 3.

599, 691, 1291, 1451, 2411, 3851, 4919, 5051, 5479, 5531, 5879, 6599, 7079, 7691, 8011, 8039, 11491, 13291, 14011, 15091, 15971, 16651, 17359, 18731, 19211, 19531, 20731, 22651, 23971, 24611, 25639, 25679, 26251, 32051, 32359, 32531, 32771, 32971, 35879, 37039, 37571, 38011, 38371

Offset: 1

Author

Jianing Song, Jun 20 2025

Keywords

• nonn
• easy

Comments

Primes p == 3 (mod 4) are precisely the rational primes in the ring of Gaussian integers.

5 is a quadratic residue of integers modulo p for p being a term of this sequence. (See A385165).

Examples

			5479 is a term since (2+-i)^125081 == 1 (mod 5479), and 125081 is divisible by neither 2 nor 3.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A385165, A385179, A385219 (the actual multiplicative orders).

this sequence < A385169 < A385180 < A385167 < intersection of A122869 and A385168, where Ax < Ay means that Ax is a subsequence of Ay.

Also a subsequence of A385191.

Programs

PARI

ord(p) = my(d = divisors((p+1)*znorder(Mod(5, p)))); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return(d[i]))) \\ for a prime p == 3 (mod 4), returns ord(2+-i, p)
isA385188(p) = isprime(p) && p%4==3 && ord(p)%2 && ord(p)%3