This looks reasonable...(2)

Euclid's well-known formula for generating Pythagorean triples, ( $a = m^2 - n^2, b = 2mn, c = m^2 + n^2$ ) does generate every primitive triple but not every triple, e.g. it's easy to verify that in the triple $(9,12,15)$ , $c=15$ cannot be written as the sum of two squares.

If n is even then we can express it as 2a and therefore $n^{2}=4a^{2}$ . If n is odd we can express it as 2b+1 and therefore $n^{2}=4a^{2}+4a+1=4(a^{2}+a)+1$

Therefore squares are either of the form 4n or 4n+1. When adding two squares we can therefore get a number of form 4n+2, 4n+1 or 4n. Therefore no number of the form 4n+3 is the sum of two squares.

3 is a counter example. In general numbers of the form 4k+3 can't be written as the sum of two squares.