Terence Tao (@tao@mathstodon.xyz)

Ten advances in mathematics and theoretical computer science

OpenAI shares new results on long-standing open problems in mathematics and theoretical computer science, including advances in geometry, cryptography, and complexity.

OpenAI spent millions to solve this famous math problem — mathematicians are furious

Mathematics relies on a community of experts openly sharing ideas.

OpenAI's Unreleased Model Astra Solves Ten Major Open Mathematics Problems

Math is hard.

Terence Tao (@tao@mathstodon.xyz)

In basic research, such as pure mathematics, one might naively expect that the natural question to ask with regards to a given problem X in a field is "What is the answer to X?". But in many cases the more valuable question is "What can be learned from studying X?" The answer to X itself can of course be one of the things learned in this process of study; but one can learn far more useful information besides, such as * What are the main difficulties to overcome to resolve X? * What new techniques can one discover in order to solve X? * Why are existing techniques insufficient to solve the problem by itself? * How does X relate to results in prior literature? * Can one uncover new connections between X and other topics Y, Z, ...? * What are some natural related or followup questions X', X'', ... to study? Nevertheless, until recently the two questions were closely aligned, to the point where it was not really necessary to distinguish the two: the only practical route to solving a difficult problem was to first address many of the subquestions listed above. (1/3)

Przemek Chojecki | PC on Twitter / X

The Growing Map of Open Mathematical Problems.We mapped 15,000+ conjectures from UnsolvedMath to show potential links between concepts.It also shows how under formalized the frontier is (less than 10%). pic.twitter.com/nm0PCpXVPf— Przemek Chojecki | PC (@prz_chojecki) August 28, 2026

Open models in perpetual catch-up

The open-closed gap, distillation, innovation timescales, how open models win, specialized models, what’s missing, etc.

OpenAI unleashes hundreds more math results upon a field already in shock

A month after resolving one of the six biggest open problems in mathematics, the company says its new internal model has toppled heaps more—this time without the multimillion-dollar price tag

The Meaning of Open

There are a lot of misconceptions about what open means, when it is the right strategy to apply, and the fundamental tradeoffs that go…

OpenAI fought dirty on career-making math problem, says NYU mathematician | TechCrunch

There is a $1 million bounty for the first person providing a solution to the Navier-Stokes existence and smoothness problem.

Terence Tao (@tao@mathstodon.xyz)

In traditional mathematics (or "Math 1.0"), a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments. As a result, new connections and collaborations are formed, followup problems are shared, and other mathematicians (both junior and senior) are attracted to the field (including in some cases the authors themselves, if they initially came from a different subject). As a result of all this activity, the proofs naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. (1/4)

Timothy Gowers @wtgowers on Twitter / X

AI has now solved a major open problem -- one of the best known Erdos problems called the unit distance problem, one of Erdos's favourite questions and one that many mathematicians had tried.https://t.co/SD1vVPkrHR— Timothy Gowers @wtgowers (@wtgowers) May 20, 2026

OpenAI’s math breakthrough played to AI’s strengths

I tried to explain OpenAI’s solution more clearly than OpenAI did.

Terence Tao (@tao@mathstodon.xyz)

An elaboration of the previous post mentioning the bounded gaps between primes problem as an illustration of the opportunity cost of converting a fruitful problem such as this solely into a competitive benchmark. [Indeed, as I point out below, this already came close to happening back in 2013.] As with most other problems worth studying in pure mathematics, the particular bound one gets on these gaps is not of much intrinsic importance. Improving Zhang's bound of 70 million, to 246, 188, or 6 would not, in itself, have significant impact on any other mathematical problem, let alone any real-world application, except for a small number of other analytic number theory problems in which a related numerical bound would also similarly improve. Even the twin prime conjecture, which asserts that the bound can be chosen to be 2, would only unlock a small number of immediate applications if the result is taken as a "black box", although the significantly more general and quantitative prime tuples conjecture of Hardy and Littlewood would be more useful in this regard. But again: the value of this problem lies not in the numerical bound _per se_, but rather in what the quest to study this problem and improve the bound reveals. Let me first quickly review some key moments in the timeline on this problem. The Quanta articles https://www.quantamagazine.org/mathematicians-team-up-on-twin-primes-conjecture-20131119/ and https://www.quantamagazine.org/a-new-generation-of-mathematicians-pushes-prime-number-barriers-20231026/ in 2013 and 2023 respectively also do a good job of covering these events. (1/8)

You don't have to be smart if you can think clearly

When you’re on fire, problems are transparent: they’re solved simply by the act of looking at them. Even complicated layers of multiple problems can simply be glanced through like stacked panes of glass. But nobody can work that way all the time.