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?

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