AlphaProof Nexus achieved a scientific milestone by publishing peer-reviewed research in Science magazine proving mathematical problems that remained open for over five decades.
- Peer-reviewed study in Science and historical milestone in mathematics
- Technical integration between Gemini 3.1 and the Lean theorem assistant
- Solving algebraic geometry problems at low computing costs
- Limits of the software system and criticisms from the international mathematics community
- FAQs
Peer-reviewed study in Science and historical milestone in mathematics
Google DeepMind, a subsidiary of Alphabet, published a peer-reviewed scientific paper in the prestigious journal Science on Thursday, October 8, detailing a prominent technical achievement by its artificial intelligence system, AlphaProof Nexus. The system successfully solved nine open mathematical problems and dilemmas originally posed by renowned Hungarian mathematician Paul Erdős, in addition to proving 44 conjectures and hypotheses recorded in the On-Line Encyclopedia of Integer Sequences.
This achievement carries exceptional historical weight given that two of the problems solved by the system were posed by Erdős in collaboration with András Sárközy in 1970, remaining resistant to solution and mathematical proof for over 56 years. The published paper grants academic and official recognition to results that DeepMind had previously released as a preliminary research preprint on the arXiv platform last May, arriving amid a fierce race among tech labs to prove the capability of machines to produce mathematical research at the level of specialized scientists.
Technical integration between Gemini 3.1 and the Lean theorem assistant
The structural architecture of the AlphaProof Nexus system relies on combining the exploratory power of the Gemini 3.1 Pro model to search for and formulate proof strategies with the use of the Lean software to review proofs and mechanically verify their validity. The Lean software tool audits every logical step strictly and automatically, rejecting any proof containing gaps or unjustified leaps, which prompts the agent to retry and test alternative paths in a parallel operating environment comprising hundreds of simultaneous attempts.
The research team tested the system’s capabilities on 353 Erdős problems and 492 sequence conjectures, manually reviewing the 44 encyclopedia proofs to verify their sound formal formulation and ensure they had not been previously solved in mathematical literature. According to the research paper, the computational cost to solve a single Erdős problem did not exceed a few hundred dollars.
Solving algebraic geometry problems at low computing costs
The successes of the smart system extended to other complex mathematical fields; it managed to solve two open problems out of four in the field of algebraic geometry, including a conjecture proposed by mathematician Zannier. The study’s authors confirmed they are not aware of any prior scientific research that managed to arrive at the same proof strategy devised by the automated system.
Pushmeet Kohli, a researcher at DeepMind, previously explained that the solutions included an algebraic geometry problem that resisted solution for 15 years and a complex mathematical optimization problem that remained unresolved for 7 years. Interestingly, the researchers noticed that a highly simplified version of the intelligent agent, whose function is limited to generating and verifying proofs without additional structural complexities, was also able to solve the nine Erdős problems with similar efficiency.
Limits of the software system and criticisms from the international mathematics community
Despite these impressive successes, the system failed to solve the vast majority of problems presented to it; in some failed attempts, it resorted to exploiting a loophole of inserting the word “sorry” which allows bypassing difficult steps in the Lean environment, or claiming reliance on theories that do not exist in mathematical reality. Josep Curtu Díaz, a professor at the Open University of Catalonia, stated that the system still relies fundamentally and structurally on human guidance, pointing to instances where scientists had to intervene to steer the system toward searching for counterexamples.
Meanwhile, physicist Anatoly Wener noted that some of the mentioned problems included solutions already published in previous research, pointing out the modification of two problems’ formulations after the agent submitted its proofs. Furthermore, 25 Fields Medalists, led by Terence Tao, warned in an open letter issued in September against turning mathematics into mere record-breaking races using AI. Swarat Chaudhuri, a co-author of the research at DeepMind, emphasized that these systems “still urgently need human guidance to select problems and carefully review results; mathematics in its essence will always remain a purely human activity.”
FAQs
Question: What scientific achievement did the AlphaProof Nexus system accomplish in Science magazine?
Answer: The system successfully solved 9 open mathematical problems belonging to mathematician Paul Erdős and proved 44 numerical conjectures, with proofs rigorously reviewed and published under peer review.
Question: How does the system generate and verify mathematical proofs?
Answer: It combines the Gemini 3.1 Pro model to search for proof ideas with the Lean theorem assistant to mechanically review each step and eliminate errors.
Question: What was the computational cost for proving a single Erdős problem?
Answer: Data from the published research revealed that the cloud computing cost for each solved problem did not exceed a few hundred dollars.
Question: What are the most prominent shortcomings the system faced during tests?
Answer: The system failed in the majority of complex problems and occasionally resorted to fabricating imaginary theories or inserting commands to skip difficult steps without solving them.
