During the Summer of 2003 I started to learn about the fascinating topic of elliptic curves while participating in the Summer REU Program at the University of Utah. Besides learning a lot and having fun, I wrote a paper on finite groups on elliptic curves. You can see the paper in either .ps, .pdf or .dvi form. I wrote it for an audience of Math majors. For more information what the REU was all about see Elliptic Curves REU.
I had spoken to both graduate students and advanced undergrads about the research I did in the Summer REU when I was approached about talking in the weekly Math Colloquium for Undergraduates. Not wanting to just regurgitate the same old things, I used Neal Koblitz's method of introducing Elliptic Curves via the Congruent Number Problem.
A congruent number is a whole number n for which there exists a right triangle with rational lengthed sides whose area is n. The first few congruent numbers are: 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126 You can find out more about these numbers at this Sequences Website link.
I hope to get some pictures posted here as well as notes from the talk in the near future.