Looks good.
Last line is a bit confusing. I'd just say that if a=0 then clearly b=2^{k}, and similarly if b=0
Problem: Show that if then either a or b is zero for some integers a, b and a non-negative integer k.
My solution attempt outline:
> Assume a>0, b>0
> Consider a Pythagorean Triple of form
> Show that we cannot obtain a primitive P.T. from above ( is not odd) so the equation does not have a solution for any k
> So, we cannot have a>0, b>0
> Show that so (a>0 and b=0) OR (a=0 and b>0)
Anything wrong with this? Does anyone have a simpler solution? Thanks.