A385218 Multiplicative orders of 2+-i modulo p == 3 (mod 4) that are congruent to 2 modulo 4.

30, 4290, 3710, 3150, 20090, 164430, 21114, 22490, 59514, 43494, 244650, 65110, 819930, 932190, 1011030, 1266750, 1405410, 533830, 1864590, 135470, 2266530, 79002, 946970, 3863190, 1039890, 4952850, 170178, 566202, 6277530, 1324930, 3091690, 9397290, 214314, 5054610, 3467950, 3511090

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 congruent to 2 modulo 4.

By definition, a(n) is the multiplicative order of 2+-i modulo A385179(n).

Examples

			a(7) = 21114 since it is the multiplicative order of 5 modulo A385179(7) = 919, and it is congruent to 2 modulo 4.
		

Links

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

Crossrefs

Cf. A385165, A385179 (corresponding primes), A385217, 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)%4==2, print1(ord(p), ", ")))