Discovery & AI
Unsolved Problems in Mathematics
Also called: Open problems
- Personal interest
- Formal theory
- Working interpretation
Clearly stated questions that nobody has answered yet invite me to experiment: trying proofs, hunting for examples that break a claim and letting computers search. I enjoy the chance of finding a small but meaningful new piece of mathematics.
What is the smallest useful claim I can prove without any gaps?
Why it attracts me
An unsolved problem with an exact statement is a rare thing: a question where everyone agrees what an answer would look like. I enjoy the chance of finding a small but meaningful new piece of mathematics, judged by a standard that does not bend.
The idea
There are three main ways to make progress. You can prove the claim for every case. You can find a counterexample, which disproves it. Or you can settle part of it, such as every case up to a certain size. Computer search is powerful for the last two. But as Proof Versus Evidence reminds me, a pattern that holds in a million checked cases is still not a proof.
An example
My Agentic Solvers side quest takes the third route. It works on well-known unproven claims about numbering the dots and lines of trees, networks of dots joined by lines with no loops. For example, it certifies that every tree with up to 23 dots can be numbered in one particular way. Each tree comes with a certificate, a record of its numbering that a separately written checker program confirms. The project does not claim a proof for trees of every size, and it is still waiting for expert review. It claims a larger checked range, which is narrower and easier to audit.
Where it connects
Part of the pull is beauty. Elegant proofs and surprising patterns draw me in (Mathematical Beauty), but beauty can only guide the search; it never replaces correctness. Machine-checked proofs (Proofs a Computer Can Check) keep that honest, and AI-led discovery (AI Systems That Discover, Not Just Summarize) gets its clearest test here.
What this does not establish
Checking every case up to a certain size does not prove a claim for all sizes. The computer-checked ranges described here are still waiting for review by human experts, including a check that they are actually new.
Questions I'm still exploring
- How do I choose problems hard enough to matter but small enough to make progress on?
- When does a pattern in computer results deserve the effort of a proof?
- How can I be confident that a problem is truly still open?
Sources and further reading
- Erdős Problems, a database of problems posed by Paul Erdős — A public list of precisely stated problems, many still open.
- Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery (Cambridge University Press, 1976) — A classic account of how proofs and counterexamples push each other forward.
Working interpretation: drafted from my notes and interests for review. It is not a direct quotation, and I may still change it.