# Some Quickies

Part III of my (never-ending) series “A Means to an End” should be up soon, but I wanted to record this little proof now before I forget it.  Incidentally, these observations will be relevant to the to-be-had discussion in my upcoming posts, but for now I’ll treat it as stand-alone material.

Recall that a pair of whole numbers $(m, n)$ is said to be an isoperimetric pair if

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

This definition arises from the work of my REU students.  Specifically, a rectangular arrangement of $mn$ squares will require the smallest perimeter possible if and only if $(m, n)$ is an isoperimetric pair.

Also, if you take the expression on the right-hand side and replace $+$ with $\cdot$, we stumble across our previously celebrated “next pronic or square” function $SP(mn)$.  Something else to point out is that the formula

$SP(x) = \left\lceil\,\frac{x}{\left\lceil\,\sqrt{x}\right\rceil}\right\rceil\,\cdot\,\left\lceil\,\sqrt{x}\,\right\rceil$

presents the output as a product of its “square” or “pronic” factors.  In particular

$\left\lceil\,\sqrt{x}\,\right\rceil =\left\lceil\,\frac{x}{\left\lceil\,\sqrt{x}\right\rceil}\right\rceil \text{ or }\left\lceil\,\frac{x}{\left\lceil\,\sqrt{x}\right\rceil}\right\rceil + 1$

These observations are helpful in proving the following neat, little facts.

Proposition 1: All pairs of the form $(n, n+2)$ are isoperimetric.  Moreover, whenever $n > 1$, all pairs of the form $(n, n+3)$ are also isoperimetric.

Proof (sketch): For the first claim, one uses the fact that $SP(n(n+2)) = (n+1)^2$ for all $n \geq 1$.  This fact itself is easily verified; high-school algebra confirms that $n(n+1) < n(n+2) < (n+1)^2$.  That is, numbers of the form $n(n+2)$ are always sandwiched in between the pronic $n(n+1)$ and the square $(n+1)^2$.  by definition of $SP(x)$, it follows that $SP(n(n+2)) = (n+1)^2$.  Using the remark above, one readily verifies that $(n, n+2)$ is isoperimetric.

For the second claim, one uses the fact that $n > 1$ to conclude that $(n+1)^2 < n(n+3) < (n+1)(n+2)$.  This immediately implies that $SP(n(n+3)) = (n+1)(n+2)$.  Comparing both sides of our isoperimetric pair equation, we indeed confirm that $n + (n+3) = (n+1) + (n+2)$.  Therefore $(n, n+3)$ is isoperimetric.  $\square$

Corollary 1: Every number of the form $k^2 - 1$ is hydropronic.

Proof: Note that $k^2-1 = (k-1)(k+1)$.  The pair of factors $(k-1)$ and $(k+1)$ are isoperimetric since they differ by $2$.  $\square$.

Corollary 2: For $k>2$ every number of the form $k(k+1) - 2$ is hydropronic.

Proof: Again, we show that such numbers have isoperimetric factors.  This follows since $k(k+1) - 2 = (k-1)(k+2)$ and these factors differ by $3$.  $\square$.

These corollaries appear to be the tip of the iceberg.  My REU students have the following conjectures that, if true, generalizes these results:

Conjecture: Every number of the form $n^2 - k^2$ is hydropronic (provided $k$ is not too big).  Also, every number of the form $n(n+1) - k(k+1)$ is hydropronic (provided $k$ is not too big).  Finally, all hydropronics are of these forms.

I should point out that $k$ being “not too big” is an interesting condition that is somewhat difficult (at the moment) to make precise.  The bottom line is that provided $n^2 - k^2$ is not less than the previous pronic $n(n-1)$, you’re good (and a similar statement holds for the other case).

Interesting, no?