Another Quickie

Picking up where my previous quickie post left off (which itself picks up where a few other posts left off) …

Let $A$ be an arbitrary hydropronic number.

Case I: $SP(A) = k^2$.

This is equivalent to stating $k(k-1) < A \leq k^2$, and it tells us that

$\displaystyle \left\lceil\,\frac{A}{\left\lceil\,\sqrt{A}\,\right\rceil}\,\right\rceil = \left\lceil\,\sqrt{A}\,\right\rceil = k$.

Moreover, the assumption that $A$ is hydropronic means that there exist factors $m$ and $n$ such that

$\displaystyle m + n =\left\lceil\,\frac{A}{\left\lceil\,\sqrt{A}\,\right\rceil}\,\right\rceil + \left\lceil\,\sqrt{A}\,\right\rceil = 2k.$

We use the above equation to solve for $m$ in terms of $n$ as $m = 2k - n$.  In turn this implies $A = mn = (2k-n)n = 2kn - n^2$.  Let us now consider the difference $SP(A) - A = k^2 - (2kn - n^2) = k^2 - 2kn + n^2 = (n-k)^2$.  In particular, we have just proven the following

If $A$ is a hydropronic such that $SP(A)$ is square, then $A = SP(A) - q^2$, i.e. $A$ is a difference of squares.

Case II: $SP(A) = k(k-1)$.

Similar reasoning in this case allows one to conclude the following:

If $A$ is a hydropronic such that $SP(A)$ is pronic, then $A = SP(A) - q(q-1)$, i.e. $A$ is a difference of pronics.

Taken together, Cases I and II $\Rightarrow$

Every hyrdropronic is either a perfect square, a pronic number, a difference of squares, or a difference of pronics.

This raises a natural question: in between a “consecutive” pair of square and pronics — say in between $k^2$ and $k(k-1)$ — there are several numbers of the form $k^2 - a^2$, but are all of them hydropronic?

Let’s explore this briefly.  One observation is that $k^2-a^2 = (k-a)(k+a)$, and we are assuming that $k(k-1) < k^2 - a^2 < k^2$.  This is only true if $a^2 < \sqrt{k} \iff a < k^{1/4}$.  (Does this look familiar, Manny?)  Under these hypotheses is it true that $(k-a)$ and $(k+a)$ are an isoperimetric pair?  Indeed it is!

Similar reasoning holds for differences of pronics.

Combined with the results above we have the following result:

If $SP(A)$ is a perfect square, then $A$ is hydropronic if and only if $A$ is of the form $SP(A) - q^2$ for some $q$ (with the possibility that $q = 0$). If $SP(A)$ is pronic, then $A$ is hydropronic if and only if $A$ is of the form $SP(A) - q(q-1)$ for some $a$ (with the possibility that $q = 1$).

Every non-square, non pronic hydropronic number is a difference of pronics or a difference of squares.

More precisely, in between $k^2$ and $k(k-1)$ all hydropronics are of the form $k^2 - a^2$, and in between $k(k-1)$ and $(k-1)^2$, all hydropronics are of the form $k(k-1) - a(a-1)$.

The REU students are busy working on exactly how large the subtracted square or pronic can be.  More specifically, they are working on a formula for this quantity, but for now we can say, “as large as possible,” which is just a way of saying that the difference needs to stay in the desired interval.

Pretty cool, huh?