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

Let be an arbitrary hydropronic number.

**Case I**: .

This is equivalent to stating , and it tells us that

.

Moreover, the assumption that is hydropronic means that there exist factors and such that

We use the above equation to solve for in terms of as . In turn this implies . Let us now consider the difference . In particular, we have just proven the following

*If is a hydropronic such that is square, then , i.e. is a difference of squares.*

**Case II**: .

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

*If is a hydropronic such that is pronic, then , i.e. is a difference of pronics.*

Taken together, Cases I and II

*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 and — there are several numbers of the form , but are *all* of them hydropronic?

Let’s explore this briefly. One observation is that , and we are assuming that . This is only true if . (Does this look familiar, Manny?) Under these hypotheses is it true that and 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 is a perfect square, then is hydropronic if and only if is of the form for some (with the possibility that ). If is pronic, then is hydropronic if and only if is of the form for some (with the possibility that ).*

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

More precisely, in between and all hydropronics are of the form , and in between and , all hydropronics are of the form .

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?