Prove that If a prime number p can be written as a sum of 2

Prove that, If a prime number, p, can be written as a sum of 2 squares, then p = 2 or p = 4n+1.

Solution

2=1^2+1^2

Hence, p=2 can be written as sum of 2 squares

Any other prime is odd of the form

4n+3 or 4n+1

We know that square of any natural number modulo 4 is either 0 or 1

Hence sum of squares of two natural numbers can be either 0, 1 or 2 modulo 4

But, p is prime and odd so it cn only be 1 modulo 4

Hence proved.