So I'm a high school student, and I am working on a paper which requires at least university level mathematics. I researched a lot online but yet still got nothing in my head.
I would like to do something about Algebra / Number Theory, and also maybe a little bit Analysis. I have a good knowledge of Galois Theory and a bit Algebraic Number Theory (i.e. Ideal, unique factorization...) but I really can't think of anything to do. I wanted to do something original (maybe a generalization) but achievable. Right now, I have brainstormed a few ideas (if you suggests new ideas that would be great, but you can also give suggestion to my ideas):
1. Diophantine $m$-tuples and Elliptic curve
I think I have read something about diophantine $m$-tuple (i.e. A set of numbers $(a_1,a_2,\cdots,a_i)$ such that $a_ia_j+1$ is a perfect square for any $i<j$) and elliptic curve. There's a conjecture about this topic whether a diophantine quintuple exists (But I don't think I can solve it).
2. Generalization of Hilbert Theorem 90
I know that Hilbert Theorem 90 can be used to solve some diophantine equation, for example the Pythagorean triples and the Pell's equation. Therefore I'm thinking if any generalization can be made on it so that it could solve more diophantine equations.
That's all for my ideas right now. If you want you could leave a comment or an answer giving me suggestions on my topics above, which would be extremely helpful for me. If you have any more new ideas that would be even better. Thanks so much for helping me. :D