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

Author

Jianing Song, Jun 20 2025

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

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

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