#16 Pronic, oblong, rectangular numbers.... Etymology and History of Math Terms

   Pronic Numbers are numbers that are the product of two consecutive integers; 2, 6, 12, 20, 30... They are also called rectangular or oblong numbers. Pronic seems to be a misspelling of promic, from the Greek promekes, for rectangular, oblate or oblong. Neither pronic nor promic seems to appear in most modern dictionaries. Richard Guy pointed out to the Hyacinthos newsgroup that pronic had been used by Euler in series one, volume fifteen of his Opera, so the mathematical use of the "n" form has a long history.

Oblong is from the Latin ob (excessive) + longus (long). The word oblong is also commonly used as an alternate name for a rectangle. In his translation of Euclid's "Elements", Sir Thomas Heath translates the Greek word eteromhkes[hetero mekes - literally "different lengths"] in Book one, Definition 22 as oblong. . "Of Quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right angled but not equilateral..."

The pronic numbers are a special class of oblong numbers in which the consecutive factors differ by one.

The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1, 1/2 + 1/6 + 1/12 + .... = 1

If you append 25 to any pronic number, you have a square of a number ending in 5.  225 = 15^2,  625 = 25^2, 1225 = 35^2....

And just for fun, try the infinite iterated square root of any number.  Like 

If you play around with it, you can figure out that it turns into a quadratic equation x^2 -x -n=0.  When n = the iterated value is 2, the positive solution is 2, but if you try 3, or 4, .. it seems like you get a messy irrational number.  Using four gives you about 2.5616..  but if you press on to try n = 6, you get a perfect integer, 3.  And by now you suspect that any pronic number iterated under the infinite radical expansion produces the larger of the sequential factors of n.  So 12 gives you  a positive root of 4.  

There is a reason for that, of course, it's math.  The square root of 4n+1 for any  pronic number is a rational number (or perhaps more simply, for any pronic n, 4n+1 is a perfect square).  

You can demonstrate why to yourself, or your students, by constructing four rectangles with area equal to a pronic number of units. Fit these together  two horizontal and two vertical and you are left with a single missing 1x1 grid.

Try the same thing with iterated cube roots, (or higher roots, there is a nice pattern) see if you can find the pattern of the rational results, and their seeds.