I’m a senior in high school, and I’ve been lucky enough to be exposed early on to some pretty advanced math. A teacher mentioned that all primes that are 1 mod 4 can be expressed as the sum of two squares, but didn’t pose it to our class as a problem to tackle. Either way, I began to tackle some parts of it I found interesting. Nobody in my class had any idea whether the problem was even in striking distance or not.
Recently, my friend showed me a brilliant piece of math that proved a big chunk of the theorem. I was able to provide one of two major lemmas required to complete the proof. All we have left to prove is that all irreducible numbers over the Gaussian Integers are prime.
We each felt like we did the easy part of the work so far, which I feel is the sign of a good collaboration. It’s totally possible this last part will be almost impossible for us, but it certainly doesn’t seem that way right now.
This experience has made me really excited to continue doing math. I won’t get to be useful to the world at large by studying math, but at least I’ll get to make some other mathematicians pretty happy.