# OpenAI's Astra Solved 10 Decades-Old Math Problems For Just $2,000

_Saturday, August 1, 2026 at 8:00 AM EDT · Science · Latest · Tier 2 — Notable_

![OpenAI's Astra Solved 10 Decades-Old Math Problems For Just $2,000 — Primary](https://imageio.forbes.com/specials-images/imageserve/64ac98a90ef5570e89534a0b/0x0.jpg?format=jpg&height=900&width=1600&fit=bounds)

OpenAI said its new Astra model solved ten long-standing mathematical problems at a reported cost of $2,000 in API calls. The company published the work on August 1 as a 249-page manuscript collection with machine-checkable certificates for every result formalized in Lean.

The problems span group theory, high-dimensional geometry, coding theory, quantum complexity, lattice cryptography, and extremal combinatorics. Each had been open for at least ten years. They include a construction establishing the existence of non-sofic groups, a disproof of Connes's rigidity conjecture, an improvement to the general upper bound on sphere-packing density in high dimensions that had stood since 1978, and three problems from Paul Erdős's catalogue.

Fields Medalist Tim Gowers said he would have recommended the proof for publication in a top mathematics journal without hesitation. A team of nine mathematicians, including Gowers and Noga Alon, published a companion paper explaining the proof. Thomas Bloom, who maintains the Erdős problem catalogue, called the August results "big news" and said they were even more significant than an earlier unit distance result. OpenAI researcher Noam Brown noted no Millennium Prize Problems were solved.

The Lean certificates allow anyone to download and verify the proofs mechanically, collapsing the traditional peer-review timeline. OpenAI acknowledged staff helped prepare papers and formalize arguments, and the $2,000 figure covers successful runs rather than all attempts. The model itself is not available for independent reproduction.

## Sources

- [Forbes](https://www.forbes.com/sites/jonmarkman/2026/08/03/openais-astra-solved-10-decades-old-math-problems-for-just-2000/)

---
Canonical: https://techandbusiness.org/newswire/TCi1DYdu9H89MW4fOQD3nY
Retrieved: 2026-08-03T17:44:32.504Z
Publisher: Tech & Business (techandbusiness.org)