Does All Odd Exist?

As $a$ is odd, we know that $a^2$ is odd too, so, if $b$ is odd too, $a^2+b^2$ is the sum of two odd numbers, which is even. Therefore there are no pythagorean triplets with only odd numbers

My way of doing it: If a and b are odd, let a=2n+1 and b=2m+1 where n and m are nonnegative integers. a^2+b^2=(2n+1)^2+(2m+1)^2 . =4n^2+4n+1+4m^2+4m+1 . =2[2n^2+2n+2m^2+2m+1] and this must clearly be an even integer.