Ten problems in advanced mathematics had resisted the specialists who study them for a living. One, a question in group theory, had stood unanswered since 1999. Each is the kind of problem a mathematician spends a career trying to close. An unreleased OpenAI model called Astra worked through all ten and published the proofs earlier this month. On the Fields Medal scale, any single one of these would plausibly anchor a medal case, one widely shared assessment on X said, after putting the question to an AI system. The Fields Medal is the highest honor in mathematics, awarded every four years for the field’s best work.

The ten problems span geometry, coding theory, group theory, quantum complexity and cryptography, subjects far from each other but united by one trait: each carries a precise answer specialists agree on, once someone finds it. Two of the results close out three entries on a catalogue of open problems left behind by the mathematician Paul Erdős. Finding the solutions cost OpenAI under $2,000 in computing, by the company’s own account. Specialists had already spent years, in some cases decades, chasing the same answers.

Each proof arrived with a file the model generated to check its own logic step by step. Mathematicians outside the project still need time to confirm the results hold up under the kind of scrutiny peer review provides. Noam Brown, an OpenAI researcher, wrote on X: the effort also came up short elsewhere. None of the seven Millennium Prize Problems, considered the hardest open questions in mathematics, gave way. He added the team had not spent much computing power on any single problem, and pushing further was possible, a sign the same approach might still reach harder targets.

Every step in these proofs gets checked against a fixed rule: it either passes or fails. This same quality, a checkable right answer, is what has made AI so much better at writing code over the past couple of years. Developers train these systems in stages, and the refining stage is where the difference shows up. One method relies on a person comparing two answers and picking the better one, a judgment call shaped by taste and opinion. The other relies on a fixed test a machine runs on its own, with no person needed to weigh in. A line of code either runs correctly or it doesn’t. A mathematical proof either holds up or it doesn’t. Problems with this kind of clear right answer are exactly where the second method excels, which is why coding and now advanced mathematics have moved so fast.

Mathematicians who track this catalogue of problems called the results big news, even as they start the slower work of checking them. The reaction echoes what happened in software development a couple of years ago, when coders who spent their days translating instructions into working code watched a machine start doing the same task. Some described a kind of unease: a skill they had built an identity around was slipping away from them. People whose work centers on a checkable answer, in math or in code, now spend their time choosing which problems the machine should tackle next, rather than working out the answers by hand. Even those celebrating the results are still working out what the solutions are worth, since resolving an abstract question in group theory does not by itself explain what changes for anyone outside the field.

Not every problem works this way. Judgment calls without a clear right answer, the kind involved in weighing a good decision or a fair outcome, sit outside what the method reaches, no matter how much computing power gets thrown at them. The proofs published earlier this month still await the slower confirmation mathematics runs on, and the hardest problems in the field remain untouched. On the narrower set of questions a machine checks for itself, the gap between a decades-old puzzle and a solved one is closing fast.

Sources

OpenAI, Ten advances in mathematics and theoretical computer science — openai.com
X, Vaibhav Sisinty on the Astra results — x.com
X, Noam Brown on the Astra results — x.com
Tech Times, OpenAI’s Astra Solves Ten Decade-Old Math Problems With Machine-Checkable Lean Proofs — techtimes.com