Kurt's Undergraduate Thesis

Available in the following formats:

As an undergraduate in the University Scholars' program at Penn State University, one of my responsibilities was to write an undergraduate thesis. The title of my undergraduate thesis is "An Analysis of the Ratio (n)/n." I did most of the work on it during the Spring semester of 1994, and I completed it in May, 1994. (I forget the exact date.) The following is a brief description of my work on this thesis, and the motivation behind it.

About My Undergraduate Honors Thesis

The function, sigma

(n) stands for the sum of the factors (or divisors) of n. For example, sigma

(24)=60, since the factors of 24 are (listed smallest to largest) 1,2,3,4,6,8,12,24, and the sum of these numbers is 60.

My thesis is about the number you get when you divide sigma (n) by n itself. For example, sigma (24)/24 would be 60/24, or 5/2 (reduced to lowest terms).

Seems like an odd thing to be concerned about, doesn't it? And perhaps it is. But remember, it was just an undergraduate thesis! And besides, it was inspired by the very interesting Odd Perfect Number Problem...

The Odd Perfect Number Problem

A "perfect number" is a positive integer, we'll call it n, with the property that the sum of its proper divisors equals n itself. For example, 6 is a perfect number, since its proper divisors are 1,2,3, and 1+2+3=6. Similarly for 28: 28 has proper divisors 1,2, 4,7,14, and the sum of these numbers is 28. A couple of other known perfect numbers are 496 and 8128. (Check these yourself!)

An important observation is that these numbers are all even; indeed, all known perfect numbers are even. Even perfect numbers are pretty much understood; that is, we know how to go about finding them. Odd perfect numbers, however, are a different story.

No one has ever discovered an odd perfect number. However, no one has been able to prove that no odd perfect number exists, either. So, no one knows whether or not such a number exists. The suspicion is that there _isn't_ one, but in mathematics you need a proof, not just a feeling or a hunch. That's one of the more appealing qualities of mathematics: usually, there are right answers. But, I digress...

If n is a perfect number, then sigma (n)/n is equal to 2. That's because the sum of all the factors of n that are less than n is equal to n, and then you add the other factor (n) to that, to get the sum of all the factors of n: sigma (n)=2n. This gives us sigma (n)/n=2n/n=2. Nice, isn't it? ...well, I think it is. Who asked you, anyway?

What I originally intended to do was to show that, if n is odd, this ratio can't be 2, and thus solve the problem and become world-famous and be rich enough to have someone else write this stuff for me... but alas, I wasn't quite that brilliant. So, I decided to see what I could find out about this ratio. I didn't prove anything earth-shattering, but I still got some interesting results. Again, interesting to me; your mileage may vary.

At this point, either you'd like to find out more, or you've had quite enough of this topic. So, if you're tired of reading about my undergraduate thesis and odd perfect numbers and my attempts to talk about math without being utterly boring, then you may as well go back to my home page. On the other hand, if you're curious (or seeking an insomnia cure), then go ahead and take a look! Feel free to download my undergraduate thesis; the available formats are gif (images), LaTeX, DVI, and PostScript.

Last modified 12/21/2000
ludwick@math.temple.edu