...OK so I'm 16 and I'm hardly going to make a major contribution to the world of science. Still.

I'm looking for ideas for a mathematics investigation that either a proof of or a counterexample of would look impressive. (I'm looking for ideas, obviously not a site that just gives me a project).

Basically its for that competition I mention from time to time (The one you may remember I did something on fuzzy graphics on).

Preferably I would like it to be pure mathematics (Mechanics I may consider, Statistics - definately not).

A few examples of titles of projects that have proved to be good for the purpose I want:
- Boolean logic - The brains behind electronics (Considered mathematics rather than computers - this one basically looked at optimising boolean circuits)
-Continued fractions on spreadsheet - a new perspective - Again a mix of computers + mathematics
- An analysis of the four colour theorem the distribution of colours and the 3rd dimesional expansion
-A proof of the conjecture that perfect numbers may never be odd

- Suggestions for projects of a similar (not too similar that it looks like I have copied) nature/standard are what I would like (and what specific areas of mathematics I'll have to read up on).
-Obviously I need to wither discover, proove or disprove something.

-Matrix algebra might be interesting.

-This project would really be my back-up plan for if I don't think I'll get my programming based one finished on time (I have grand plans, however am unsure if they will materialise).
-Of course, if it might involve the usage of programming to prove something... all the better.