A372953 Orders of finite fields where -1 is a square.

2, 4, 5, 8, 9, 13, 16, 17, 25, 29, 32, 37, 41, 49, 53, 61, 64, 73, 81, 89, 97, 101, 109, 113, 121, 125, 128, 137, 149, 157, 169, 173, 181, 193, 197, 229, 233, 241, 256, 257, 269, 277, 281, 289, 293, 313, 317, 337, 349, 353, 361, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461

Offset: 1

Author

Mike Speciner, May 17 2024

Keywords

• nonn
• easy

Comments

The sequence comprises the positive powers of 2, the positive powers of primes congruent to 1 mod 4, and the positive even powers of primes congruent to 3 mod 4.

The multiplication group of GF(p^n) is cyclic of order o = p^n-1. For p=2, 1=-1, so 1 is a square root of -1. Otherwise, -1 has order 2 and so any square root of -1 has order 4. So, for there to be a square root of -1, o mod 4 must be 0, i.e. p^n mod 4 = 1. Then if g is a generator of the group, g^(o/4) is a square root of -1. p^n mod 4 = 1 if and only if p mod 4 = 1 or p mod 4 = 3 and n is even.

Crossrefs

Cf. A000040 (primes), A000961 (prime powers).

Symmetric difference of A000079 (power of 2) and A085759 (prime powers congruent to 1 mod 4).

Programs

Python

from itertools import count
from msmath.numfuns import primepower
def a(start=2,stop=None) :
  for n in range(start,stop) if stop else count(start):
    if primepower(n) :
      if n%4 != 3: yield n