The Total Tokens Needed to Find Solutions Would Cost Roughly $2,000 at Sol API Rates

OpenAI announced ten new results in mathematics and theoretical computer science, each resolving or making substantial progress on a long-standing open problem. The results were achieved by an internal version of Astra, its next major model, and span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. The total tokens needed to find solutions would cost roughly $2,000 at Sol API rates.

Humans prepared manuscripts with the same model, and each argument was formalized in a Lean certificate. OpenAI also released a model's narration of its thinking process for each solution. The company acknowledged concerns about AI's role in mathematics, including the Leiden declaration, and stated that claiming human authorship for AI-generated proofs would misrepresent contributions.

It takes responsibility for correctness while noting the mathematical arguments were generated by its system. The announcement follows a May disproof of the Erdős unit-distance conjecture and an initiative providing 100,000 scientists and mathematicians with free access to ChatGPT for Academic Researchers.