A385217 Odd multiplicative orders of 2+-i modulo primes p == 3 (mod 4).

13695, 40755, 7475, 19895, 43995, 117855, 138075, 13185, 69445, 87725, 308505, 220665, 567645, 80735, 1103355, 1321125, 1386945, 507795, 1594005, 130995, 205975, 2051325, 2092035, 2216565, 2703975, 1368315, 2750685, 504095, 3039345, 212605, 3342405, 125081, 1274665, 3991725, 152205, 4279275

Offset: 1

Author

Jianing Song, Jun 22 2025

Keywords

• nonn
• easy

Comments

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

Odd elements in A385165.

By definition, a(n) is the multiplicative order of 2+-i modulo A385169(n).

Examples

			a(49) = 635253 since it is the multiplicative order of 5 modulo A385169(49) = 8731, and it is odd.
a(50) = 57447 since it is the multiplicative order of 5 modulo A385169(50) = 8839, and it is odd.
a(51) = 2029953 since it is the multiplicative order of 5 modulo A385169(51) = 9011, and it is odd.
		

Links

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

Crossrefs

Cf. A385165, A385169 (corresponding primes), A385218, A385219.

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)
forprime(p=3, 1e4, if(p%4==3 && ord(p)%2, print1(ord(p), ", ")))