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 is said to be an isoperimetric pair if
This definition arises from the work of my REU students. Specifically, a rectangular arrangement of squares will require the smallest perimeter possible if and only if is an isoperimetric pair.
Also, if you take the expression on the right-hand side and replace with , we stumble across our previously celebrated “next pronic or square” function . Something else to point out is that the formula
presents the output as a product of its “square” or “pronic” factors. In particular
These observations are helpful in proving the following neat, little facts.
Proposition 1: All pairs of the form are isoperimetric. Moreover, whenever , all pairs of the form are also isoperimetric.
Proof (sketch): For the first claim, one uses the fact that for all . This fact itself is easily verified; high-school algebra confirms that . That is, numbers of the form are always sandwiched in between the pronic and the square . by definition of , it follows that . Using the remark above, one readily verifies that is isoperimetric.
For the second claim, one uses the fact that to conclude that . This immediately implies that . Comparing both sides of our isoperimetric pair equation, we indeed confirm that . Therefore is isoperimetric.
Corollary 1: Every number of the form is hydropronic.
Proof: Note that . The pair of factors and are isoperimetric since they differ by . .
Corollary 2: For every number of the form is hydropronic.
Proof: Again, we show that such numbers have isoperimetric factors. This follows since and these factors differ by . .
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 is hydropronic (provided is not too big). Also, every number of the form is hydropronic (provided is not too big). Finally, all hydropronics are of these forms.
I should point out that being “not too big” is an interesting condition that is somewhat difficult (at the moment) to make precise. The bottom line is that provided is not less than the previous pronic , you’re good (and a similar statement holds for the other case).