I'm in the middle of preparing a new course that will be teaching +Python and +Sage Mathematical Software System to our first year undegraduates at Cardiff university. As such I'd appreciate as many great example as I can get so please do let me know of anything :)
I use the term mathematics in a loose sense to include 'things' that are not necessarily proofs:
- identifying conjectures;
I'm familiar with the 'well known' results that can be found on this +Wikipedia page: http://goo.gl/EjyKel
I can also recommend the book 'A=B' by Petkovsek, Wilf and Zeilberger which you can download here: http://www.math.upenn.edu/~wilf/AeqB.
The search for prime numbers is also a pretty cool thing that I can talk about.
Even some example of things that have an obvious need of computers would be great to hear about. For example I'm quite familiar with discrete event simulation techniques that I'll be mentioning to the students. Here's a video I put together a while back showing how to use a computer to simulate a queue: