In Julian Havil's The Irrationals, he at one point in Chapter 2 supposedly states a proof that there are no points with rational coordinates on the curve $x^2+y^2=3$. He begins:
"Assume that a rational point $(p/q, r/s)$ lies on the circle and so $p^2/q^2+r^2/s^2 = 3$, resulting in $p^2s^2+q^2r^2=3q^2s^2$ and so $(ps)^2+(qr)^2 = 3(qs)^2$ and therefore the existence of positive integers so that $a^2+b^2=3c^2$. Since the sum of two even numbers and the sum of two odd numbers are each even, it must be that one of $a^2$ and $b^2$ is even and the other odd; so..."
and continues. I understand every step other than this - why does the fact that one of $a^2$ and $b^2$ must be even and the other odd from the simple fact about the sums of odd and even numbers?
