OpenAI's Astra AI system solves 10 long-standing mathematical problems at $2,000 per solution, validating advanced reasoning compute ROI.

Something unusual is happening in the world of pure mathematics. OpenAI has announced that an internal AI system has helped crack ten mathematical and computer-science problems that have remained unsolved for at least a decade, and in many cases considerably longer. The announcement, paired with a separate initiative providing advanced ChatGPT access to 100,000 researchers, signals that AI advances in mathematics are moving from novelty to genuine research infrastructure.

OpenAI's approach prioritizes access over closed development. The company recently launched ChatGPT for Academic Researchers, providing 100,000 scientists and mathematicians free access to its most capable ChatGPT models. The rationale is straightforward: access has historically been the bottleneck for AI-assisted research. Academic budgets rarely accommodate premium AI subscriptions at scale, and few institutions can fund individual researchers testing frontier models on niche mathematical questions. By removing that barrier for a large cohort of scientists, OpenAI is betting that wider access will surface results like those it now showcases.

In May, OpenAI shared an early signal of this momentum: an AI-generated disproof of the Erdős unit-distance conjecture, a decades-old problem attributed to mathematician Paul Erdős. The disproof surfaced almost incidentally during evaluation of an unreleased model rather than through a dedicated research campaign. What makes this significant is that the result has already inspired follow-on work from human mathematicians, with subsequent papers building on the approach. When an AI-generated result triggers further research, it suggests these systems contribute extendable techniques rather than isolated answers.

The centerpiece of OpenAI's announcement is a batch of ten new results addressing problems that had seen no progress on their core questions for at least ten years, many considerably longer. These problems span eight distinct mathematical disciplines: high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. That breadth is notable—these are not adjacent sub-questions in a single niche but distinct open problems from separate corners of mathematics and computer science, each with decades-long research histories.

All ten results were produced by Astra, OpenAI's next-generation internal model. The critical detail is cost: OpenAI estimates that total token usage to find solutions to all ten problems would amount to roughly $2,000 at API rates. For research on which human mathematicians made no progress over a decade or more, a few thousand dollars in compute represents a striking price point and signals how AI is shifting the economics of tackling hard open problems.

The process combined human oversight with AI capability. Human researchers took the model's raw arguments and developed them into structured manuscripts while working alongside the same model. The model then formalized each argument into a Lean certificate—a machine-checkable proof format used to verify mathematical claims rigorously. OpenAI also released a narration of the model's reasoning process for each solution, providing outside researchers visibility into how the system arrived at each answer.

This workflow—AI-generated arguments, human-prepared manuscripts, and machine-formalized Lean certificates—establishes a division of labor where AI systems generate mathematical insight while formal verification tools and human oversight ensure confirmation. This template could prove influential as more AI-assisted results emerge across other open problems.

The broader implication extends well beyond ten solved problems. If a single internal model can produce publishable-grade results across eight mathematical disciplines for roughly $2,000 in compute, the constraint on tackling long-standing open problems shifts from mathematical insight toward access to capable models—exactly what OpenAI's academic access program expands. This combination of inexpensive computation and wide researcher access will likely draw attention from rival AI labs and from mathematics departments integrating these tools into their work.

The disproof of the Erdős conjecture has already spurred additional research exploring related questions in sum-product theory, incidence geometry, and computational complexity. Whether this momentum continues at the current pace, or whether other AI labs produce comparable results with different models, will likely determine how quickly AI-assisted proof generation becomes routine in mathematical practice.