{
  "id": 13412687,
  "title": "OpenAI's Astra solved 10 decades-old math problems for $2,000",
  "url": "https://urgent.news/2026/10/10/openais-astra-solved-10-decades-old-math-problems-for-2-000",
  "topic": "ai",
  "section": "AI",
  "published": "2026-10-10T13:00:00.000Z",
  "source": {
    "name": "Dev.to",
    "slug": "dev-to",
    "url": "https://dev.to/ayraix/openais-astra-solved-10-decades-old-math-problems-for-2000-3a88"
  },
  "original_language": "en",
  "account": "On August 1, 2026, OpenAI announced that its undisclosed internal model, named Astra, had successfully generated fully-verified proofs for ten previously unsolved mathematical problems and a theoretical computer science challenge. The model achieved this feat for a mere $2,000 in computational resources, a fraction of the cost of a weekend's worth of GPU rental. The impressive results were documented in a 249-page manuscript and the accompanying Lean 4 proof certificates, which were made publicly available under an Apache 2.0 license on GitHub. Lean, a proof assistant, meticulously checked every logical step, ensuring the correctness of the proofs. No gaps were left unchecked, as the repository's \"sorry\" count (indicating unproven gaps) remained at zero across all ten results. The problems tackled by Astra span various mathematical domains, including non-sofic groups, sphere-packing bounds, Connes' rigidity conjecture, Ehrhart's volume conjecture, and several problems from Paul Erdős's open-problem catalog, such as problem 183 on multicolor Ramsey numbers. The $2,000 figure underscores the remarkable efficiency of Astra, which essentially condensed years of human effort into an overnight computation using machine-speed proof search. While initial reactions to the results were positive, particularly from mathematician Timothy Gowers, the proofs have not yet undergone formal peer review, and thus, remain subject to community validation. This breakthrough marks a significant milestone in demonstrating the potential of AI models to independently generate valid mathematical proofs, underscoring the growing influence of formal proof checkers in establishing the veracity of mathematical claims.",
  "summary": "The headline number: on August 1, 2026, OpenAI said an internal, unreleased model it calls Astra produced fully machine-verified proofs for ten open problems in mathematics and theoretical computer science — several unsolved for decades — for a total inference cost of roughly $2,000. That is less than a weekend of GPU rental, spent on questions professional mathematicians had not cracked in up to…",
  "key_points": [
    "OpenAI's undisclosed model Astra solved 10 unsolved math problems",
    "Astra achieved this for $2,000 in computational resources",
    "Proofs verified by Lean 4, no gaps found in 249-page manuscript"
  ],
  "editors_take": null,
  "illustration": null,
  "coverage": {
    "outlets": 2,
    "also_reported_by": [
      {
        "outlet": "NDTV - India News",
        "title": "OpenAI's Math Papers Destroyed Careers Of Early Mathematicians: US Professor",
        "url": "https://urgent.news/2026/10/10/openais-math-papers-destroyed-careers-of-early-mathematicians-us",
        "published": "2026-10-10T07:06:58.000Z"
      }
    ]
  },
  "ai_generated": true,
  "disclaimer": "Summaries, key points and the editor’s take are written by software from other outlets’ reporting and may contain errors — always check the linked original."
}