If I know that $S$ is a square but not its square root, how do I find the next square number? Given a square number $s = n^2$, is there a way to find the next square $s'$ without knowing $n$? (That is, if you can't take the square root of $s$ to determine $n$, can you compute $s'$?)
 A: Well, there's at least no real rational function that does what you want.
Suppose $f(n^2)=(n+1)^2=n^2+2n+1$ for all $n\in\mathbb N$ (or even just for an infinite number of $n$). Then $\frac{f(n^2)-n^2-1}{2}=n$, so the rational function $\frac{f(x)-x-1}{2}$ intersects $\sqrt x$ an infinite number of times, which is impossible, since rational functions are asymptotically equivalent to polynomials, and $\sqrt x$ is either infinitesimal to any particular polynomial or the other way around (iff the polynomial is constant).
So if it's possible at all, it's going to take more than just multiplying, adding and dividing.
A: I assume that all you wanted was not taking any square roots- we could actually determine n without doing that.
Suppose $s=n^2$, then write $s-1=n^2-1=(n+1)(n-1)$. Given any such factorisation (ie. $s-1=(m-1)(m+1)$ for some m), we must have m = n as we require $m^2-1=s-1=n^2-1$. Hence the n obtained in the factorisation is unique, and we are done.
eg. Given $s=49$, write $s-1=48=8*6$ and your $n$ must be $7$.
A: A semi-answer perhaps.
$(n+1)^2 = (e^{\frac{ln(n^2)}{2}}+1)^2$
Technically I did not take a square root, however I did implicitly compute $n$ by using the term $e^{\frac{ln(n^2)}{2}}$

Addendum: You can avoid ever computing $n$ by using the function$f(x) = (e^{\frac{3}{2}ln(x)}+x)^2/x^2$
$f(n^2) = (e^{\frac{3}{2}ln(n^2)}+n^2)^2/(n^2)^2= (n+1)^2$
