[imranq]

I guess you are referring to Yitang Zhang who created the first bound on the twin-prime conjecture. These days mathematics has become so specialized that its rather difficult to come up with anything new that is noteworthy and not already discovered. Just browsing through the graduate math library the other day and the stacks and stacks of journals dating from the 1700's and from various parts of the globe made me realize just how much math is out there.

Still, like Ian Stewart said in his "Letters to a Young Mathematician", math is an inverted pyramid - when a problem is solved, it leads to more branches to be solved. So maybe breakthroughs are possible from a determined amateur.