A385219 Multiplicative orders of 2+-i modulo p == 3 (mod 4) that are not divisible by 2 or 3.

7475, 19895, 69445, 87725, 80735, 205975, 504095, 212605, 125081, 1274665, 720055, 181445, 1044005, 492929, 891335, 1346365, 5501795, 7360445, 8179505, 9489095, 10628035, 3850775, 3138905, 14618765, 15377605, 34181, 17907265, 21377825, 23942035, 5047511, 13694965, 6868865, 28713125

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.

Elements in A385165 that are not divisible by 2 or 3.

By definition, a(n) is the multiplicative order of 2+-i modulo A385188(n).

Links

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

Crossrefs

Cf. A385165, A385188 (corresponding primes), A385217, A385218.

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 && ord(p)%3, print1(ord(p), ", ")))

Formula

a(9) = 125081 since it is the multiplicative order of 5 modulo A385188(9) = 5479, and it is divisible by neither 2 nor 3.