Find an explicit formula for a square integer between $4n^3$ and $4n^3 + 4n^2$

Question

Given any positive integer $n>1$, it is fairly straightforward to prove that there exists at least one (and perhaps many) square integers lying strictly between $4n^3$ and $4n^3 + 4n^2$. (Proof: This is equivalent to the claim that there is an integer between $2n\sqrt{n}$ and $2n\sqrt{n+1}$, which is true because $2n\sqrt{n+1} - 2n\sqrt{n} > 1$.) Does anybody see a way to find an explicit formula (in terms of $n$, preferably as the square of a polynomial) for such a square integer?

Answer 1

Let's suppose we have a polynomial $r$ such that $r(n)^2$ is a perfect square with $$4n^3 \le r(n)^2 \le 4n^3+4n^2$$ for all $n$. Suppose $r(x)$ has degree $k$. Then $$ \frac{4n^3}{n^{2k}} \le \frac{r(n)^2}{n^{2k}} \le \frac{4n^3}{n^{2k}}+\frac{4n^2}{n^{2k}}.$$ Letting $n$ go to infinity, the middle expression approaches a non-zero constant, and we conclude that $2k\ge3$ and $2k\le3$, i.e., $k=\frac{3}{2}$. Since $k$ is an integer, this is a contradiction. Hence, no such polynomial $r$ exists.