A385165 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 2+-i modulo p in Gaussian integers.

8, 48, 30, 180, 528, 96, 1848, 2208, 1740, 1496, 360, 1560, 2296, 10608, 11448, 5376, 4290, 1932, 11400, 8856, 27888, 16020, 1216, 3300, 3710, 49728, 51528, 14280, 3150, 69168, 7344, 80088, 8568, 48360, 13695, 40136, 6444, 44896, 7980, 146688, 29260, 92880, 48180

Offset

1,1

Comments

A002145 are precisely the rational primes in the ring of Gaussian integers.

From the representation of complex numbers as 2 X 2 matrices, a(n) is also the multiplicative order of the matrix [2,-1;1,2] or [2,1;-1,2] modulo p.

a(n) is divisible by ord(5,p): If (2+-i)^n == 1 (mod p), then 5^n == 1 (mod p).

a(n) divides (p+1) * ord(5,p), since we have (2+-i)^(p+1) == 5 (mod p).

If 5 is a quadratic residue modulo p, then ord(5,p) divides (p-1)/2, and so a(n) divides (p^2-1)/2. Conversely, if a(n) divides (p^2-1)/2, then (x+-y*i)^2 == 2+-i (mod p) for some integers x, y, and so (x^2+y^2)^2 == 5 (mod p), which means that 5 is a quadratic residue modulo p.

Links

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

Example

The multiplicative order of 2+-i modulo A002145(3) = 11 is a(3) = 30, since (2+-i)^30 == 1 (mod 11), and 30 is the smallest such exponent.

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, 1e3, if(p%4==3, print1(ord(p), ", ")))

Crossrefs

Cf. A002145, A211241, A385163 (multiplicative order of 1+-i), A385166.

Sequence in context: A024108 A392755 A247726 * A263506 A216323 A335351

Adjacent sequences: A385162 A385163 A385164 * A385166 A385167 A385168

Keyword

nonn,easy

Author

Jianing Song, Jun 20 2025

Status

approved