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

1,1

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).

Links

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

Example

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

Prog

(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), ", ")))

Crossrefs

Cf. A385165, A385179 (corresponding primes), A385217, A385219.

Sequence in context: A166845 A166834 A246620 * A216728 A196466 A159401

Adjacent sequences: A385215 A385216 A385217 * A385219 A385220 A385221

Keyword

nonn,easy

Author

Jianing Song, Jun 22 2025

Status

approved