clopen.solutions
Problems932
Open574
Solved358

305 of the solved problems have a claimed solution not yet confirmed. Of the open problems, 51 have a partial result.

Problems by status over time

About clopen.solutions

Large language models are making rapid progress in mathematics, and the traditional venues are struggling to keep up. As a result, many new results are scattered across GitHub repositories, personal blogs and other sites, and it has become hard to tell whether a question is still open or has been closed. In a sense, it is clopen. The aim of clopen.solutions is to give an up-to-date overview of the progress on open problems, collected in one place.

How it works

Besides its title, every problem has its own permanent identifier, such as OP-00001, much like an arXiv number or a DOI. To keep the registry up to date, a number of GitHub repositories are checked every other day for work that might resolve a problem. Papers can also be submitted here directly, including partial results. To prevent abuse, every submission is reviewed by a frontier AI model before it is announced.

Where the problems come from

Many problems are drawn from lists that others have assembled, and each problem's page names its source. We are grateful to the people who maintain them:

Others come straight from the papers that pose them.

Questions

What counts as an open problem here?

A question posed in the literature that, as far as the curators know, nobody has answered. Each one has a source, a paper or a published list, and a statement short enough to read in a minute, with the background in its references.

What do the statuses mean?
Open
No claim to resolve the problem is on record.
Partially resolved
A paper on record settles part of the problem, such as a special case or a weaker bound.
Solution claimed
A paper on record claims to resolve the problem, but no journal publication or independent check of it is on record yet.
Solved
A claimed resolution has been accepted by a journal, or checked independently, and a curator has verified the record.
Lean verified
A Lean proof has been checked by machine, and the theorem it proves has been confirmed to be the problem as stated here.
Needs verification
A curator has flagged the record for a closer look, for instance because of an objection to a claimed solution.
Withdrawn
The problem is no longer listed as a problem, for instance because it turned out to be ill-posed or a duplicate. Its identifier stays, so links to it keep working.
What does the AI-Verified badge mean?

Before a paper submitted here is announced, two AI referees review it: one reads the argument, the other checks that the paper addresses the problems it claims. If they find at most minor objections, the paper gets the AI-Verified badge and their report is published with it; serious objections go to a curator first. The badge belongs to the paper, not the problem: it never changes a problem's status, and it is not a proof of correctness.

Papers that curators record from elsewhere, such as those collected from an approved GitHub repository, may be listed before they are reviewed. They carry no badge until a review finds none but minor objections.

Does a status mean a proof is correct?

No. A status reports the evidence on record, such as a journal's acceptance or a Lean check. It is not an endorsement, and nobody guarantees that a paper is correct.

How do I submit a solution?

Create an account with your email address and confirm it. You can then submit a paper and say which problems it addresses, whether it resolves them or makes partial progress. Its LaTeX sources are built here, and a Lean proof, if you attach one, is checked here too. Nothing is public until it is announced: by you, or for you once the automated review gives it the AI-Verified badge, if you asked for that when you started it.

My paper is already on arXiv. Can it be listed?

Yes. A paper on arXiv, Hexagon or Zenodo, or in a GitHub repository the registry has approved, can be linked rather than uploaded, by one of its authors or by a curator as a record of someone else's work. Its page links to where it is published. The approved repositories are checked every other day, and new papers in them are recorded. Authors who would rather not have their work listed can write to contact@clopen.solutions.

How do I cite a problem?

By its identifier and address, which never change, for example: OP-00001, clopen.solutions, https://clopen.solutions/OP-00001/. Please cite the problem's original source as well; its references are on its page.

I found a mistake, or a problem has been solved. Who do I tell?

Write to contact@clopen.solutions. A problem that has been solved elsewhere is the error we most want to hear about.