A385163 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 1+-i modulo p in Gaussian integers.

8, 24, 40, 72, 88, 40, 56, 184, 232, 264, 280, 312, 328, 408, 424, 56, 520, 552, 120, 648, 664, 712, 760, 792, 840, 296, 904, 952, 200, 1048, 1080, 376, 408, 1240, 120, 1384, 1432, 1464, 1512, 1528, 1672, 344, 584, 1768, 1848, 1864, 1912, 1944, 1960, 664, 2008, 2088, 2184, 2248, 456

Offset: 1

Author

Jianing Song, Jun 20 2025

Keywords

• nonn
• easy

Comments

Also, a(n) is the multiplicative order of the matrix [1,-1;1,1] or [1,1;-1,1] modulo p.

Note that (1+-i)^4 = -4. Since (1+-i)^n is a real number if and only if n is divisible by 4, we have a(n) = 4*ord(-4,p), where ord(a,p) is the multiplicative order of a modulo p.

Examples

			For A002145(4) = 19: Since (1+i)^(4k) = (-4)^k, we have (1+i)^72 == 1 (mod 19), and 72 is the smallest such exponent. Hence a(4) = 72.
		

Links

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

Crossrefs

Cf. A002145, A384164 ({a(n)/8}), A385165 (multiplicative order of 2+-i).

Programs

PARI

forprime(p=3, 1e3, if(p%4==3, print1(4*znorder(Mod(-4,p)), ", ")))