How were these problems selected?

Problems were selected to be of inherent interest to mathematicians, and to have resisted serious attempts to solve them. AI progress on them can therefore be interpreted as advancing human mathematical knowledge.

Problems were proposed for inclusion in the benchmark by working mathematicians. Our editorial board then reviewed the submissions for mathematical notability, with the minimum bar being the “Moderately Interesting” tier, characterized below.

Problems are only included if their solutions can be verified computationally. This lets us evaluate AI systems at scale without requiring humans to read AI-generated proofs each time.

The editorial board consists of Thomas Bloom, Daniel Litt, and Dan Romik. You can read their commentary on the project here.

What do the notability tiers mean?

Problems are categorized into one of four tiers, roughly capturing the significance of a solution.

Moderately Interesting: The problem was posed at least 10 years ago and has been worked on by at least two independent teams of mathematicians. It has connections to broader questions, but is not a priority for any large research program.

Solid Result: A strong researcher in the area would be happy if their median output addressed problems of this caliber. Still, the problem would probably not get much engagement outside of its subfield.

Major Advance: The median person working in a broad area of mathematics (on the scale of number theory or graph theory) would take note, and would likely make the time to understand at least the outline of the solution.

Breakthrough: The median mathematician would want to know about this result, even if it was outside their area. It would be a candidate for one of the best results of the year in all of mathematics.

When multiple tiers seemed plausible for a problem, we erred in the conservative direction. It would be disappointing to downgrade a problem’s notability after it was solved, whereas we can always highlight any unexpectedly interesting elements of a solution.

How can you verify an answer if the problem is currently unsolved?

While most novel math results are conveyed via difficult-to-verify natural language proofs, some results can be verified computationally. This generally takes the form of constructing precisely-specified objects, where brute force search for the desired object is intractable but verification is still tractable.

Each FM:OP problem thus comes with a custom verification program, or “verifier”. In some cases the verifiers are very simple, like checking an integer solution to a polynomial equation. In other cases they are sophisticated, involving specialized mathematical computing software.

We allow verifiers that provide strong numerical evidence that the AI system has solved the problem, without constituting full proof. For instance, factoring certain integers over a certain size in under a certain amount of time is believed to be impossible without a general mathematical breakthrough. It is easy to verify if an AI system can factor such integers in such time limits, but observing such instances does not constitute proof of a general result.

Allowing such problems greatly expands the scope of the benchmark. We aimed to tune the verifiers so that it is essentially certain that a positive result means the AI solution is somehow interesting, though we expect some additional tuning will be required over time.

Does the verifiability constraint introduce a bias to the kinds of math represented here?

Yes. The problems have a constructive bent, asking for concrete mathematical objects or algorithms. As a result, very many important questions in mathematics cannot be included in the benchmark. It may even turn out that AI is especially well-suited to this sort of verifiable problem.

In terms of mathematical areas, the benchmark skews toward combinatorics and number theory, though more discrete topics in other areas — e.g. knot theory in topology — are also represented.

What happens if AI solves a problem?

We inform relevant mathematicians, including those who proposed the problem, so that those who are interested can help exposit the solution. We also retire the solved problem from the benchmark, as future AI solutions won’t reflect a novel discovery. This benchmark is thus somewhat unusual in that we do not report percentage scores for individual AI systems.

What happens if a human solves a problem?

We remove it from the benchmark. In fact, this happened once during the development of the problem set: the core problem solved in this paper had been accepted for inclusion in the benchmark but was removed when the human solution was published.

Are all of the problems solvable?

Probably not. Often only one “side” of a statement is computationally verifiable. For instance, we can verify that a given mathematical object satisfies some sought-after properties, but we cannot verify a claim that no such object exists. Furthermore, some such objects may exist but be too large to verify computationally. Based on aggregating very rough guesses about each problem, we estimate that a significant proportion are not solvable: very likely at least 10% and probably (hopefully) less than 40%.

We excluded problems where the mathematical consensus is that no solution exists, e.g. asking for a counterexample to the Riemann Hypothesis. But we did not exclude cases where there is not much consensus either way, especially if the problem is significant. For instance, we include a problem that amounts to disproving the smooth Poincaré conjecture in four dimensions.

If a problem is shown to be unsolvable through conventional means — e.g., it is proved that a sought-after object does not exist, and that proof is accepted by the mathematical community — we will update the benchmark accordingly. The problem will be marked as solved by AI if the resolution was primarily AI-generated, or removed if the resolution was primarily due to human effort.

Why didn’t you use Lean for verifiability?

We had a few reasons for this, though none of them should be interpreted as suggesting that Lean-based benchmarks aren’t also valuable. Given recent advances in AI formalization capabilities, we expect formal methods like Lean to play an increasing role in AI benchmarking.

As for our reasons, first, we wanted the freedom to include problems which cannot be easily formalized in Lean today. Second, we wanted the benchmark to separate mathematical ability from formalization ability. In particular, it is very hard to predict how substantial an effort would be required to formalize the solution to any given problem, when we don’t know what such a solution might include. Third, Lean is still maturing as a programming platform, and likely has a higher potential for inscrutable bugs which AI systems might exploit.

Do you test AI systems on these problems?

Yes, we test frontier systems around when they are released. We typically test flagship releases from Anthropic, OpenAI, and Google, as well as other systems at our discretion. If a verifier accepts an AI solution, we record that status on the individual problem pages. Once a solution has been confirmed by a mathematician, we increment the headline chart.

At the time of launch (July, 2026), our scaffold includes a Python tool, a command line tool, and a web search tool. Models are given a one-million token output budget per sample, and compact as necessary. We run models on each problem four times. As of release, we hope to conduct some elicitation experiments — e.g., copying techniques used in recent high-profile AI math discoveries — and plan to share our results and open-source any code we think might be useful.

I have a solution to one of the problems! What should I do?

Email math@epoch.ai and we will do our best to run the relevant verifier in a timely manner. Please make sure your solution is formatted precisely as requested by the prompt on the problem page.

Note that we cannot adjudicate claims to show that a problem is unsolvable. We rely on typical processes within the mathematical community to settle such claims.

Who has access to the problem verifiers?

Access to the verifiers is available for purchase by any party. Proceeds help defray the cost of creating the benchmark. Our main cost is compensation to mathematicians, as the problems and verifiers are labor-intensive to formulate and implement.

As of the project launch (July, 2026), OpenAI is the only entity to have purchased access to the verifiers. OpenAI funded the creation of the original FrontierMath: Tiers 1-4, but FrontierMath: Open Problems is developed independently and owned solely by Epoch.

Email math@epoch.ai to inquire about purchasing access to the verifiers.

How do you know the verifiers work correctly?

We test the verifiers on variants of the open problems where solutions are known. We also conduct typical code reviews, including AI code review using frontier models.

How do you handle problems about breaking existing records, after the initial record is broken?

We mark such problems as “solved” once the initial record is broken. This is because we only intend to include problems where significant effort has been devoted to breaking the current record. If the initial success is due to AI finding a simple improvement that was overlooked, we will clearly describe that in the solution update on the problem page. In many cases, it will be reasonable — from the point of view of math research — to simply set the new record as a new target to break again. However, the new record typically won’t have received as much attention as the prior record — which makes the significance of breaking it somewhat harder to understand. By default, our plan is to continue to test AI systems on the new targets but not to consider them official benchmark entries any longer.