First of all, I would just like to tell everyone that you should go to stumbleupon.com and sign up for this great application to your tool bar. You sign up and then you click on all of your interests and then you will have a "Stumble" button in the top left corner of your browser and whenever you click it you will "stumble upon" some very interesting and entertaining websites which you would never have found if you were just looking on google and what not.

The reason I brought that up was because while I was "stumbling" I came across this website which talked about perfect numbers and how Fermat and a many other well known mathematicians contributed to the theory of perfect numbers. A perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself), or σ(n) = 2n. It goes on to say how Fermat's little theorem was basically discovered in his invesigation of perfect numbers. "Shortly after writing this letter to Mersenne, Fermat wrote to Frenicle de Bessy on 18 October 1640. In this letter he gave a generalization of results in the earlier letter stating the result now known as Fermat's Little Theorem which shows that for any prime p and an integer a not divisible by p, ap-1- 1 is divisible by p. Certainly Fermat found his Little Theorem as a consequence of his investigations into perfect numbers."…"Using special cases of his Little Theorem, Fermat was able to disprove two of Cataldi's claims in his June 1640 letter to Mersenne." It then goes on to talk about how Mersenne became interested and that is how he came up with the "Mersenne primes."

I think this was very interesting because sometimes we don't know where theorems may have come from and why we want to know certain things and it's amazing how many of the mathematicians and theorems we have talked about in this class were all part of one investigation of perfect numbers. It's amazing how some of these things tie together. Some of this website is still way over my head but I thought it was cool that I could relate. Maybe you guys can get even more into it if you are that interested in stuff like this…

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html