I think this would make an interesting project. The Brent, Cohen, and te Riele paper
(reference at http://www.utm.edu/research/primes/references/refs.cgi/BCR91)
sketches a proof that any odd perfect number must have at least 8 distinct factors and at least 300 digits. Furthermore, their method of proving this would have discovered such an odd perfect number if one had existed with less than 300 digits. It would be interesting to see how much this bound can be improved with the increase in computing power since then. Since GIMPS can be looked at as the search for even perfect numbers, a search for odd perfect numbers seems a natural complementary project. The beauty of these papers on odd perfect numbers is that they are not extremely technical, and I think most amateur number theorists with only minimal background would be able to understand them.
