Let $k$ be strictly bigger than $1$. Is there any integer $k$ such that $(4k-1)^2+(4k)^2$ is a perfect square?
My computation shows that there are infinitely many such $k$, namely those arising from the Pell equation $X^2-2Y^2=-1$.
However the solution manual says that there are no such $k$.
Could anyone tell me the right answer and solution preferrably without using Pell equation?