I am not a math major but I would bet that it is VERY unlikely for you to make any kind of break through in any field with only your four years of university level education. Why do you think the people who make the breakthroughs spend 4-6 years getting their PhD first?
My guess would be something computer science related. One thing about these sorts of problems is that a lot of things aren't so much "pure math' but engineering problems that involve a good deal of trial and error.