The Next Square or Pronic

REU students Emmanuel Daring, Isabel Guadarrama, Samantha Sprague, and Carrie Winterer — perhaps better known via their awesome blog — have been working hard putting the final touches on their proof of the “Lego Isoperimetric problem.”

Unlike the classical case, this “single” problem actually consists of two distinct questions.  The first of these isoperimetric problems I’ll label as IP1:

IP1:  What is the largest amount of area one can enclose using a fixed number of bricks?

The second problem goes like this:

IP2: What is the smallest number of bricks one can use to enclose a given area?

Answers to IP1 were determined by a previous group of REU students who worked with Dr. Alan Jamieson and me during the summer of 2010.  Much of their time went into formulating the problem so that it was both interesting and tractable, and this year’s team did a fabulous job of picking up where they left off.

Isabel, Manny, Samantha, and Carrie have determined that the least perimeter $P$ needed to enclose a fixed amount of area $A$ is … well, I’ll let you read their precise formula over at their own blog.  I only want to discuss one small piece of their formula, a function I’ve decided to label $SP(x)$.

Given a real number $x$, let $SP(x)$ denote the smallest square or pronic number that is greater than (or equal to) $x$.  That is

$SP(x) = \min \left\{y : y \geq x, \, y = a^2 \text{ or } y = a(a+1)\right\}$

So, for instance, $SP(2) = 4$ while $SP(5) = 6$.  As an exercise, you might want to check that $SP(180) = 182$.  After much trial and error and lots of thinking and guessing, Samantha Sprague had a key insight in expressing $SP(x)$ as a formula.  She determined that

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

This turns out to be correct (although the group is currently putting together a proof of this formula) and raises an interesting question: are there any other ways to express $SP(x)$?

The defining components of $SP(x)$ are useful for making other definitions, too.  We will say (if only temporarily) a number $N$ is hydropronic if $N$ can be factored as $N = cd$ where

$c + d = \left\lceil\frac{N}{\lceil\sqrt{N}\,\rceil}\right\rceil + \lceil\sqrt{N}\,\rceil$

From this definition, one easily determines that all perfect squares are hydropronic.  With a little work one can also verify that all pronics are hydropronic.  Unfortunately, it remains unclear what else can be said.  This formula should enable us/one to determine additional structure (if any) the set of hydropronics enjoys, but more work is needed.  Interestingly, this formula can be rephrased in more geometric terms, ones that make it clear no prime number greater than 3 is hydropronic.  I am not yet certain how easily that conclusion is made using the above criteria.

In any event, three cheers for $SP(x)$, an aptly named function as it always returns a Square or Pronic but also because it was our own Samantha SPrague who first glimpsed it.