Conjectures

Tracking down references from The Subtle Art of the Mathematical Conjecture, I ran across Art of Problem Solving, which bills itself as providing online courses ranging from "Introductory Math (Prealgebra to Geometry)" to "Advanced Math (Calculus to Olympiad Geometry)". It looks like an interesting resource: Twin Primes Conjecture, which leads to Brun's Constant (≈ 1.90216058) which might be a little heavy going for most advanced high schoolers. After the fact you can follow the proof, but coming up with it yourself would be challenging.

Anyhow, the Quanta link at the top is worth reading.

Mathematicians noodle around and find some pattern that looks interesting, and might be generally true. Then--you make a conjecture: X is true. Try to see if it is. If it is, publish. Or not, when the reviewer reminds you that it is a simple corollary of Bzirp's Theorem, published in 1905 in a Czech-language journal. (You have to know the jargon well to do a thorough literature search, and I'm not as well-versed as the pros. No, that didn't happen to me.)

Sometimes I've found my conjecture to be false, sometimes true and trivial, and most frequently I can't figure it out. The interesting conjectures are the ones that are simple, seem to be pretty easy, and take years--because the problem is really deep. Fermat's famous theorem was finally proved, but IIRC the journey there involved details about properties of elliptic curves and ran to hundreds of pages and several branches of math. I'm not sure he could have put it in the margin even with microfilm.