Prove that if x and y are odd natural numbers, then $x^2+y^2$ is never a perfect square. Prove that if x and y are odd natural numbers, then $x^2+y^2$ is never a perfect square.
Let $x=2m+1$ and $y=2l+1$ where m,l are integers.
$x^2+y^2=(2m+1)^2+(2l+1)^2=4(m^2+m+l^2+l)+2$
Where do I go from here?

You can now look at all the natural numbers modulo $4$. We know that numbers must either be even or odd, hence they have the form
$$2n\text{ or } 2n+1.$$
In modulo $4$, they are just $2n\text{ or } 2n+1\mod4$. Now, look at the squares of these numbers, we have that 
$$(2n)^2\equiv 4n^2\equiv0\mod4.$$
Also, the odds give
$$(2n+1)^2\equiv 4n^2+4n+1\equiv1\mod4.$$
Now we know that squares MUST be either $0$ or $1$ modulo $4$. What is your number modulo $4$?
