Primes that have factors in the complex plane

Fermat's Two Square Theroem

An odd prime number p can be expressed as a sum of two squares if and only if p ≡ 1 (mod 4) ie one more that a number divisible by 4.

Here are some examples:

5 = (4 X 1) + 1 = 2² + 1²
13 = (4 X 3) + 1 = 3² + 2²
17 = (4 X 4) +1 = 4² + 1²

These p ≡ 1 (mod 4) primes are not the only prime numbers the rest are of the form p ≡ 3 (mod 4) ie three more than a number divisible by 4.

Here are some examples:

7 , 11 , 19 , 23 These p ≡ 3 (mod 4) cannot be expressed as the sum of two squares.

Proofs of the theorem are very complex as shown at this link

Here is a video that includes a visual explanation which is more easy to follow though still somewhat complex:

Here is the video that first introduced me to the idea: https://youtu.be/EV-v-9KbJ7Y