### The Joy of Homogeneity, a Sequel

In the previous post, The Joy of Homogeneity, I followed Gary Davis in establishing a statement observe by Ben Vitale. Ben's observation had to do with fractions in which both the numerator and denominator were sums of consecutive odd numbers. So that, for example,

and, more generally,

Allen Pinkall left a commenet on the original page with an observation on a regularity in a somewhat modified fraction:

where the denominator starts with the last term of the numerator and not with the next one. The new observation concerns the sum of the numerator and the denominator of the reduced fraction. Let

and

Now then , where and . What we note is that the sum is twice the last term in the numerator (as well as the first term in the numerator.) This is always true. Furthermore, the fraction is irreduceable unless is even. For even, the fraction can be further reduced by just a factor of :

such that the sum of the numerator and the denominator of the latter is , exactly the last term of the sum in the numerator (as well as the first term in the numerator.)

Thanks for showing your proof! Much more succinct than my use of 4i+1 and 4i+3.

I also noticed the evens have a similar property which can be shown in the same way.

Thanks again for your blog!

Allen

March 14th, 2012 at 3:10 pmAllen, I'd give you credit for the observation if I knew your last name.

March 14th, 2012 at 11:20 pmMy name is Allen Pinkall. I've enjoyed following your blog, but I've realized I don't know much about you either. Do you have a bio page?

March 15th, 2012 at 7:49 amThank you, Allen. You can find a little bit at

http://www.cut-the-knot.org/wanted.shtml

March 15th, 2012 at 9:23 am