When AI Does Math
Mathematics has always been the purest form of human reasoning. For centuries, theorems were proven by human minds working through logic, step by careful step. The idea that a machine could contribute to mathematical discovery felt like science fiction.
That changed in May 2026, when OpenAI shared an AI-generated disproof of the Erdős unit-distance conjecture—a problem in discrete geometry that had resisted solution for decades. Now, just three months later, they have gone much further.
Ten Problems, Ten Solutions
OpenAI recently announced results from their internal version of Astra, their next major model. The model produced new results for ten problems that have been open for at least a decade, and in most cases much longer. These are not minor improvements—they are substantive breakthroughs across multiple fields:
| Problem | Field | Result |
|---|---|---|
| High-dimensional sphere packing | Geometry | New upper bounds on sphere-packing density |
| Binary and spherical codes | Coding theory | Exponentially improved bounds |
| Non-sofic groups | Group theory | Construction establishing existence |
| Connes rigidity conjecture | Operator algebras | Disproof of longstanding conjecture |
| Arithmetic circuit complexity | Complexity theory | New lower bounds for computing permanent |
| Quantum parallel repetition | Quantum complexity | Exponential parallel repetition theorem |
| Closest vector problem | Lattice cryptography | Polynomial-factor hardness of approximation |
| Ehrhart volume conjecture | Convex geometry | Determined maximum volume in every dimension |
| Multicolor Ramsey numbers | Extremal combinatorics | Superexponential lower bound |
| Extremal number conjectures | Graph theory | Results on compactness and degeneracy |
The problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics.
The Cost of Discovery
Here is a number worth noting: the total computational cost to find solutions to all ten problems would be roughly $2,000 at API rates. For context, some of these problems have been open for over fifty years. The human effort spent on them—by some of the brightest mathematicians alive—cannot be calculated in dollars, but it spans careers.
This is not to diminish human mathematicians. The model arguments were prepared into manuscripts by humans, and the model formalized each argument in Lean—a proof assistant that mechanically verifies mathematical reasoning. The collaboration between AI and human verification is what makes these results trustworthy.
What This Means
For the mathematical community, these results are both exciting and unsettling. Exciting because long-stuck problems are finally moving. Unsettling because the nature of mathematical proof is changing.
OpenAI addresses this directly in their announcement:
“We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system contribution and the nature of genuine human intellectual work.”
This is a refreshing honesty. The company generated the mathematical arguments, helped prepare the manuscripts, and formalized the proofs. They take responsibility for correctness while being transparent about the source.
The Broader Picture
This announcement comes alongside ChatGPT for Academic Researchers, providing 100,000 scientists and mathematicians with free access to ChatGPT. The strategy is clear: position AI as a research collaborator, not a replacement.
The May disproof of the Erdős conjecture has already inspired follow-up research from mathematicians worldwide. Five subsequent papers have built on that initial result. This is how mathematics progresses—one breakthrough opens doors to many more.
A New Era
We are witnessing something unprecedented. AI systems are no longer just checking arithmetic or computing integrals. They are generating novel mathematical arguments, finding patterns that humans missed, and solving problems that have resisted attack for generations.
The question is no longer whether AI can do mathematics. It is how quickly the field will adapt, and what new questions will emerge when the bottleneck shifts from “can we prove this” to “what should we ask next.”
What does it mean for mathematics when the hardest part becomes asking the right questions?