Credit: WaveGenerics; AI-generated
Introduction
Anyone following recent developments in artificial intelligence (AI) recognizes that these models have advanced rather dramatically in the past year, perhaps most remarkably in mathematical prowess. Here is a brief summary of some recent developments; see this previous Math Scholar article for additional details.
- Counterexample to an Erdős conjecture. An OpenAI model found a counterexample to a graph theory conjecture first proposed by legendary mathematician Paul Erdős in 1946 (see here for more details).
- High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
- Binary and spherical codes. Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
- Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
- Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.
- Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
- Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
- Solution of a famous Navier-Stokes conjecture. In September 2026, OpenAI announced that it had solved a famous conjecture relating to the Navier-Stokes equations, which govern fluid flow. This conjecture has been listed as one of the ten Millennium Prizes by the Clay Mathematics Institute. See here for more details.
Professional research mathematicians have gasped at these developments, notably the last item (the Navier-Stokes conjecture, which by the way has generated some controversy). How far can these AI models go? What will OpenAI and other organizations announce next?
OpenAI releases 722 new papers
On 6 October 2026, mathematicians found out: OpenAI released a collection of 722 papers, covering 377 different problems, all produced by OpenAI’s AI models. Some caveats: Only part of these results have been certified by Lean or similar proof-checking software, and mathematicians have not yet had time to determine definitively which of these results are incremental improvements to results already in the literature, and which represent major advances. Even so, the list of these results is nothing short of stunning. Here are just a handful of highlights, gleaned by the present author, in number theory, algebra and complex analysis:
- The irrationality exponent of $\pi$ is $2$. This resolves a decades-old conjecture about $\pi$. In particular, the OpenAI model claims to have shown that for every $h > 0$ and all sufficiently large denominators $q$, every rational $p/q$ satisfies $|\pi – p/q| \geq q^{-2-h}$. The previous best result, due to Zeilberger and Zudilin (2019), had $q^{-7.103-h}$.
- Catalan’s constant is irrational. OpenAI claims to have proven that Catalan’s constant, namely $G = \sum_{k \geq 0} (-1)^k / (2k+1)^2 = 0.915965594\ldots$, is irrational. This conjecture is at least 70 years old; a 2013 paper described Catalan as the most basic constant for which irrationality is suspected but not proven.
- Quasi-Riemann hypothesis results. The Riemann hypothesis, namely the assertion that the nontrivial zeroes of the Riemann zeta function $\zeta(z)$ all lie along the line ${\rm Re}(z) = 1/2$, has arguably reigned as the most important unanswered question in mathematics for over 100 years. While this remains unproven, OpenAI was able to prove some weaker assertions, including that $\zeta(z)$ has no zeroes in the strip $7/8 \leq {\rm Re}(z) \leq 1$.
- Hilbert’s tenth problem over rational numbers. OpenAI’s model proved that no algorithm can decide whether a polynomial with integer coefficients has a rational zero.
- The Abelian Zilber-Pink conjecture. The model proved that every irreducible subvariety of an Abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.
- Squarefree quartiles and power-free polynomial values. The model proved that every irreducible integer quartic with no fixed prime-square divisor assumes squarefree values with the predicted positive Euler-product density.
- Quadratic bound for Jacobsthal’s function. Let $h(k)$ be the least integer such as that interval of $h(k)$ consecutive integers contains an integer relatively prime to any given positive integer with at most $k$ distinct prime divisors.
- Short Egyptian fractions. OpenAI’s model proved a conjecture of Erdős: For every sufficiently large integer $b$, every rational $a/b$ with $1 \leq a < b$ is a sum of $O(\log \log b)$ distinct positive unit fractions, with an absolute implied constant.
- Modularity of elliptic curves over imaginary quadratic fields. The model proved that every elliptic curve over every imaginary quadratic field is modular, with matching local parameters.
Full details provided by OpenAI on these and many other results are available here.
We should emphasize that the above are just a few items in the areas of number theory, algebra and complex analysis. There are hundreds more, in fields such as algebraic and complex geometry, theoretical computer science, dynamical systems and ergodic theory, combinatorics, probability and statistical mechanics, mathematical logic, mathematical physics, operator algebras, topology, functional analysis, differential geometry and partial differential equations.
Indeed, the enormous scope and breadth of these results is arguably the most impressive feature of this collection. Professional research mathematicians seldom master more than a single field of specialization; OpenAI’s model has mastered 17 different areas, spanning virtually all of modern mathematics!
Reaction from the mathematical community
Needless to say, OpenAI’s latest release has stunned the mathematical community and has attracted considerable attention to the field — see New York Times, Scientific American, New Scientist, Washington Post and The Economist.
Some mathematicians have criticized organizations such as OpenAI for aggressively hiring top researchers away from universities. Others have criticized OpenAI and others for not yet fully releasing all of its documents, including the prompts its researchers used to produce the output. Others, including UCLA researcher Terence Tao, have criticized the “insane” pace of these results.
Many other mathematicians are grieving over what appears to be a major tectonic shift in their field. Indeed, many online postings are revealing combinations of grief, confusion and fear: “Mathematics will never be the same.”
Scott Aaronson of the University of Texas, Austin, lamented, “Human mathematicians are forevermore dethroned as the main theorem-proving entities on planet earth.” Prominent mathematician Ken Ono told students at U.C. Berkeley, “You might be graduating into a profession that might not even exist, or that will be very different than what you expected. … You need to brace.” Daniel Litt of the University of Toronto added, “The existing equilibrium has broken. … We’ll have to find a new one.”
Hope for the future?
Pavel Etingof of the Massachusetts Institute of Technology offers a more optimistic outlook:
Mathematicians will be spending less time trying to peck on the permafrost, getting out mathematical gems with great effort. … They’ll be sourcing that kind of pecking more to AI and trying to interpret and understand and learn the results that will come from that. There will be a lot more mathematics to make sense of. And there will be a lot of work for mathematicians — maybe different work from before, but definitely interesting work.
Jordana Depelewicz sketches one path forward:
It will be hard. Inertia is powerful. But mathematics could come out of this stronger than before. Forced to name and prioritize what people love and value about the field, current and future mathematicians can emphasize skills that have previously been deprioritized: the ability to communicate and share knowledge, to develop a broad vision, to experiment with original ideas without feeling the need to publish theorem after theorem. To think deeply.
Either way, we are in for “interesting” times. Hold on to your hats!