16. If a positive integer cannot be written as the difference of two square numbers, then the integer is called a "cute" integer. For example, 1,2 and 4 are the first three "cute" integers. Find the $2010^{\text {th }}$ "cute" integer.
(Note: A square number is the square of a positive integer. As an illustration, 1,4,9 and 16 are the first four square numbers.)
16. Answer: 8030
Any odd number greater than 1 can be written as $2 k+1$, where $2 k+1=(k+1)^{2}-k^{2}$. Hence all odd integers greater than 1 are not "cute" integers. Also, since $4 m=(m+1)^{2}-(m-1)^{2}$, so that all integers of the form $4 m$, where $m>1$, are not "cute". We claim that all integers of the form $4 m+2$ are "cute". Suppose $4 m+2$ (for $m \geq 1$ ) is not "cute", then
$$
4 m+2=x^{2}-y^{2}=(x-y)(x+y)
$$
for some integers positive integers $x$ and $y$. However, $x+y$ and $x-y$ have the same parity, so that $x-y$ and $x+y$ must all be even since $4 m+2$ is even. Hence $4 m+2$ is divisible by 4 , which is absurd. Thus, $4 m+2$ is "cute". The first few "cute" integers are $1,2,4,6,10 \cdots$. For $n>3$, the $n^{\text {th }}$ "cute" integer is $4 n-10$. Thus, the $2010^{\text {th }}$ "cute" integer is $4(2010)-10=8030$.