Also, generally "prodigy" means that, as a child, one was far beyond one's age group. If you're learning algebra at 8 and calculus at 11, you're a prodigy..

That doesn't feel sufficient to me. I usually interpret the word to imply achieving high levels of status while still a child (for example, winning national competitions, touring internationally as a perfomer, etc.). Merely learning stuff won't do that.