DeepMind's AlphaProof Nexus Published in Science, Solves 9 Erdős Open Problems

According to QbitAI reporting, the research results of Google DeepMind's AI mathematical proof framework AlphaProof Nexus appeared in Science and completed peer review. It is a Gemini-driven AI mathematical proof agent framework. The report also mentions that the paper has 20 authors in total, among whom Aja Huang was also a core researcher on AlphaGo back then.

The evidence boundary must be drawn first: the above information, as well as all problem-solving counts and cost figures involved in this article, come from the retelling of a single source, QbitAI. No first-party original data from DeepMind or Science was provided, and independent third-party verification is lacking. Therefore, whenever specific results and costs are involved below, they are treated as source claims rather than confirmed facts.

The report claims that AlphaProof Nexus solved 9 open Erdős problems, two of which had been open for 56 years. In addition to the Erdős problems, it also proved 44 open conjectures in the OEIS encyclopedia of integer sequences, solved a 15-year-old open problem in algebraic geometry concerning the log-concavity of Hilbert functions, and improved a long-standing theoretical bound in convex optimization for an anchored gradient descent method.

These numbers form the core of the external narrative for this achievement, but they currently remain at the level of source claims. The report does not give conclusions from independent review by the mathematics community, nor does it state whether these proofs have been accepted by the academic community.

The report gives relatively specific descriptions of three problems. Erdős #12 was proposed in 1970 and, by the time of the report, had been unresolved for 56 years. It asks whether one can construct an infinite set of integers that satisfies strict divisibility constraints and is sufficiently large. The report says its solution is a block construction: first split the infinite set into integer blocks, use the Chinese remainder theorem together with the property of sets avoiding three-term arithmetic progressions to satisfy local divisibility conditions, and then reassemble the complete infinite set.

Erdős #125 was proposed in 1996 and had no settled answer for 30 years. The answer given by AlphaProof Nexus is that the lower density is zero, and the report says the key to the proof is that log₄/log₃ is irrational. Erdős #846 was proposed in 1992 and had blocked humans for 34 years. The report says it completed the proof by mapping the edges of an infinite complete graph to points in the plane and combining this with infinitary combinatorics.

These three summaries are the report's retelling of the proof ideas. The specific proof details are given only by that report, with no first-party materials attached. They can illustrate the coverage of the results, but they are not enough to support an independent judgment about the correctness of the proofs.

The report claims that the core working method of AlphaProof Nexus is that Gemini 3.1 Pro writes proofs, Lean checks them, and error feedback is returned to Gemini for revision and rechecking until all proofs pass. Around this workflow, DeepMind designed four agent configurations.

Agent A has multiple independent sub-agents trying in parallel. Each sub-agent generates Lean code through Gemini and then continuously revises it based on compiler feedback, with no additional auxiliary tools throughout. Agent B adds AlphaProof, so that when it encounters difficult sub-problems it can call on it for targeted search. Agent C introduces evolutionary search: multiple agents place different proof drafts into a shared database, and then a scoring agent based on Gemini 3.0 Flash compares the drafts' plausibility, clarity, and novelty, aggregating them into Elo scores to select routes for further exploration. Agent D integrates AlphaProof and evolutionary search; it is the full version and also the main configuration used in this batch exploration of open problems.

The report also mentions that DeepMind selected 353 problems from those with existing Lean formalized versions and gave them to AlphaProof Nexus to attempt, and ultimately it successfully solved 9. This denominator is worth remembering: it shows that the results were obtained within a pre-formalized candidate pool.

The report claims that DeepMind's control experiment found that the simplest Agent A could also solve all 9 Erdős problems, just more expensively on hard problems. In other words, without evolutionary algorithms and without AlphaProof, a single LLM loop plus compiler feedback was enough to cover all 9 problems.

This result runs counter to the intuition that more features mean greater capability, and it directly weakens the narrative of architectural necessity for Agents B, C, and D. The difference given by the report lies in cost: on difficult problems such as #125 and #138, the full version Agent was about 2 to 5 times more cost-efficient than the basic version; all told, the cost of solving one problem was roughly a few hundred dollars.

But these two cost figures are estimates from the report, with no explanation of the calculation basis, and the benchmark and statistical method are unclear. The details and reproducibility of the control experiment itself are also unknown. Therefore, the more prudent reading is: the evidence presented by the report points to 'a simple loop was already enough to solve this batch of problems,' not 'complex architectures are useless.'

The report claims that the entire set of Lean proof code has been made public and can be directly inspected by outsiders. This is the most verifiable part of this achievement: the formal nature of Lean means that, if the code and statements are correct, the correctness of the proofs can in principle be machine-checked.

But public code does not automatically equal acceptance by the mathematics community. The report does not state whether these proofs have undergone independent review, and it remains unclear whether the 44 OEIS conjectures and the 15-year algebraic geometry problem have been accepted by the academic community. What public code solves is 'whether it can be checked,' not 'it has already been checked and endorsed.'

The report contrasts this achievement with OpenAI's release style: OpenAI dumped 722 mathematical proofs at once and met collective resistance from the mathematics community, whereas DeepMind completed peer review. The report also mentions that two mathematicians from Carnegie Mellon University and the University of South Carolina wrote a commentary in the same period in Science titled 'AI for mathematical research has arrived.'

This contrast is the report's narrative frame. The OpenAI matter appears in the article only as comparative background, with no independent source; the connection between this achievement and the dispute over OpenAI's release style also lacks direct responses from both sides. The specific content of the contemporaneous commentary is not elaborated in the source, so it should not be used to infer the mathematics community's overall attitude toward this achievement.