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.

Case 1. p=2

Then, p=1^2+1^2

HEnce, p can be 2

Case 2. p>2

So p must be odd

Square of any natural number gives 0 or 1 modulo 4

Hence is p is sum of 2 squares ie

p=x^2+y^2

So we have 3 cases

1. x^2 and y^2 both give 0 modulo 4. But that would mean x and y are both even abd hence p is even which is a contradiction

2. x^2 gives 0 and y^2 gives 1 modulo 4 or vica versa

Hence, p=x^2+y^2=0+1=1 mod 4

3. x^2 gives 1 and y^2 gives 1 modulo 4

Hence, p=1+1 =2 mod 4

Hence, p is even which is a contradiction

Hence, only possiblities

p=2 or p=1 mod 4

Hence proved