Ask HN: Can an amateur make contributions to pure math or theoretical physics?

I've always been fascinated by theoretical physics (specifically, quantum mechanics) and pure math (specifically, number theory). However, they're not subjects I'd like to make a career out of. Realistically, might I be able to make meaningful contributions do these fields as a part-time amateur? That is, someone who has no formal training beyond possibly undergraduate courses and is otherwise self-taught.