Google DeepMind says its AlphaProof Nexus system solved nine open problems posed by Paul Erdős, including two that had been unsolved for 56 years. The approach pairs a large language model (LLM) with the Lean proof assistant, which machine-verifies every step. DeepMind reports each solved Erdős problem cost only a few hundred dollars. The system also formalized 44 open conjectures from the Online Encyclopedia of Integer Sequences (OEIS).
How the system works
AlphaProof Nexus generates proof proposals with an LLM and submits them to the Lean proof assistant for formal verification. If Lean rejects a step, the LLM continues to search for an alternative until it produces a proof that the verifier accepts. According to DeepMind, this loop—LLM generation plus formal checking—enabled the solution of multiple longstanding problems at relatively low cost.
What was solved
DeepMind states that nine open Erdős problems were resolved, two of which had remained open for 56 years. In addition, the system processed and formalized 44 conjectures from the OEIS. The company emphasizes that the financial cost per Erdős problem was only a few hundred dollars.
Context and significance
Paul Erdős left a large backlog of open problems that have occupied mathematicians for decades. The DeepMind announcement followed a similar OpenAI announcement by one day. Both efforts relied on formal, machine-checked proofs, indicating a broader trend: combining LLMs with formal verifiers shifts the bottleneck from the scarcity of human insight to the existence of a verification loop that can accept an automatically generated proof.
Practical implications
DeepMind’s claim that individual proofs cost a few hundred dollars represents a marked difference from traditional, human-driven research in terms of time and resource commitments. If the method generalizes, many previously hard-to-reach open problems could be formalized and resolved at much lower cost, with implications for how mathematical research is prioritized and funded.
Parallel developments
The DeepMind release was published one day after an OpenAI announcement that made similar claims about solving Erdős problems. Both reports center on formal verification as the key enabler, suggesting multiple teams are converging on comparable techniques.
Conclusion
According to Google DeepMind, AlphaProof Nexus used an LLM together with the Lean proof assistant to produce machine-verified solutions to nine Erdős problems (including two unsolved for 56 years) and to formalize 44 OEIS conjectures, with per-problem costs of a few hundred dollars. The company highlights the importance of formal checking and the low cost of producing these proofs, a development that could change the economics and workflow of mathematical problem solving.
