If $a$ and $b$ are odd perfect squares, then $a + b$ is not a perfect square. 
Prove: If $a$ and $b$ are odd perfect squares, then $a + b$ is not a perfect square. 

Proof by Contradiction:
If $a$ and $b$ are odd perfect squares then $a = (2k+1)^2$ and $b = (2r + 1)^2$.
Assume $a + b$ is a perfect square. 
\begin{align}a + b &= (2k+1)^2 + (2r + 1)^2\\
&= 4(k^2 + r^2) + 4(k + r) + 2\\
&= 4(k + r)^2 + 4(k + r) -2kr +  2\\
&= z^2 + 2z -2kr +  2 \hspace{0.5cm} \text{where } z = 2(k+r)\\
&= (z+1)^2  -2kr +  1\\
&= (z+1)^2  - q \hspace{0.5cm} \text{where } q = 2kr + 1\end{align}
Hence, $a+b$ is in inexpressible as a perfect square when $a, b$ are odd.
Is this a valid proof? I'm not sure how to approach it any other way.

In you second line you have shown that a+b mod 4 = 2. Is this possible if a+b is a perfect square ?
