Say $$n+k^2=m^2$$for positive $m$ and $k$ integers with $m>k$. Then $$n=(m-k)(m+k)$$Keep in mind that $k$ is positive. Hence there are only $\lfloor{\dfrac{n}{2}}\rfloor$ possibilities for $m-k$, namely $$m-k=\{1,2,\cdots,\lfloor\dfrac{n}{2}\rfloor\}$$$\lfloor{\dfrac{n}{2}}\rfloor$ is always smaller than $n$ positive integer, hence even all of these possibilities gave us correct solutions for $n+k^2=m^2$, their number simply does not exceed $n$. P.S. The number of possibilities for $m-k$ is equal to the possibilities for $k$