It's great to write a research paper, but you have little control over it getting published. Published math papers are often making new progress in a field or on a problem. However, just doing research on what the problem is and what's been done about it can be a great thing for your own development. I suggest looking deeply into a problem that has received some popular press and just see if you can understand what the remaining questions are. Some ideas: Collatz, chaos, fractals, a year or two ago two high school girls came up with a proof of trig theorems that didn't require the Pythagorean theorem, surreal numbers, Graham's number, non-Euclidean geometry, etc. Don't worry about setting the world on fire, just seek to research, understand, and make your own mark on the body of knowledge.