---
title: "Google DeepMind's AlphaProof Nexus Solves 9 Open Erdős Problems: Two Waited 56 Years for a Solution"
description: "Google DeepMind's research agent AlphaProof Nexus has solved 9 open Erdős problems, two of which waited 56 years for a solution, and proved 44 OEIS conjectures with machine verification in Lean."
date: 2026-10-18T11:11:09.208Z
lang: en
url: https://xab.info/en/posts/alphaproof-nexus-google-deepmind-erdos-problems
tags: [google-deepmind, alpha-proof-nexus, artificial-intelligence, mathematics, erdos-problems, lean, formal-proofs]
author: "Alexander Buran"
publisher: "XAB.info"
---

# Google DeepMind's AlphaProof Nexus Solves 9 Open Erdős Problems: Two Waited 56 Years for a Solution

![Abstract visualization of a neural network and digital connections for an article about artificial intelligence in mathematics](https://xab.info/media/2026/10/18/alpha-proof-nexus-google-deepmind-zadachi-erdosha/alpha-proof-nexus-google-deepmind-zadachi-erdosha-1.webp)

## Key Points

- AlphaProof Nexus solved 9 out of 353 open Erdős problems, two of which went unsolved for 56 years.
- The system proved 44 out of 492 open OEIS conjectures; each Erdős problem cost several hundred dollars in computing.
- Proofs are formalized in Lean, which checks every logical step and returns feedback upon error.
- A simpler agent also solved all 9 problems, but was more expensive on hard cases — search organization matters alongside the LLM.
- The system also applies to algebraic geometry, convex optimization, graph theory, and quantum optics, but fails on tasks requiring new theory.

Google DeepMind has presented research results showing its research agent, AlphaProof Nexus, capable of solving not just Olympiad problems, but open research-level mathematical challenges. The system solved 9 out of 353 open problems associated with the Hungarian mathematician Paul Erdős, with two of them remaining unsolved for 56 years. In addition, the agent proved 44 out of 492 open conjectures from the On-Line Encyclopedia of Integer Sequences (OEIS). According to the authors, computing costs for each solved Erdős problem amounted to a few hundred dollars.

### Nine Erdős Problems and 44 OEIS Conjectures

The work focuses on using large language models to find formal proofs. The authors deliberately went beyond standard competition tasks and tested the approach on genuinely open problems that mathematics has struggled with for years. Success in 9 Erdős problems and 44 OEIS conjectures demonstrates that the approach is suitable for exploratory mathematics, rather than just educational and competitive examples. Importantly, the agent failed to solve most open Erdős problems: current successes are concentrated in areas with an existing developed library of formalized results where complex reasoning can be broken down into separate subproblems.

### Formal Verification Instead of a Convincing Explanation

The key difference between AlphaProof Nexus and a regular chatbot is its use of Lean, a language and environment for formalizing mathematical proofs. The AI proposes a proof in formal format, after which Lean checks every logical step. If the check detects an error, the agent receives feedback and continues the search. A convincingly written explanation is not enough: the result must pass machine verification according to strict logical rules. It is this "generation-verification-correction" loop that makes the process suitable for real mathematics rather than generating plausible text.

### Architecture: Multiple Agents and an Evolutionary Algorithm

In AlphaProof Nexus, multiple AI agents search for solutions, verify approaches, and refine proofs in parallel. In the most complex configuration, their work is coordinated by an evolutionary algorithm, while AlphaProof—an automated theorem-proving system developed by Google DeepMind and based on reinforcement learning—is used for specific search stages. This multi-agent architecture allows the search to be distributed across different strategies and select the most promising proof branches.

### A Simple Agent Succeeded Too — But at a Higher Cost

The authors separately compared the complex architecture with a simpler agent that simply alternates proof generation by the language model with verification in Lean. Upon re-analyzing the nine solved Erdős problems, the simpler agent also succeeded in all nine. However, on the most difficult problems, its use was more expensive. This indicates that the outcome depends not only on the capabilities of the language model itself, but also on the organization of the proof search: the complex architecture saves computation on difficult cases.

### Horizons and Limits

The system's capabilities are not limited to Erdős problems and OEIS conjectures. AlphaProof Nexus helped solve an open problem in algebraic geometry, improve bounds in convex optimization, and detect errors in the formalization of mathematical statements found in scientific literature. It is also used in graph theory and quantum optics research. At the same time, the system inherits the limitations of language models: search results can vary wildly from attempt to attempt, and problems requiring the development of extensive new theory remain beyond its capabilities. The authors believe that formal proofs can become a way to integrate AI into genuine mathematical work rather than just getting answers from it.

## Fact-Check & Sources

- [Агент Google DeepMind решил 9 открытых математических задач Эрдёша — 2 из них не могли решить 56 лет](https://www.ixbt.com/news/2026/10/17/agent-google-deepmind-resil-9-otkrytyx-matematiceskix-zadac-erdesa-2-iz-nix-ne-mogli-resit-56-let.html) - Primary source: 9 out of 353 Erdős problems, 2 unsolved for 56 years, 44 out of 492 OEIS conjectures, hundreds of dollars in cost, Lean's role, and multi-agent architecture.
- [ИИ-агент Google DeepMind решил 9 задач Эрдёша, две ждали 56 лет](https://itzine.ru/news/ai/ii-agent-google-deepmind-reshil-9-zadach-erdyosha.html) - Matches on numbers (9/353, 44/492, 56 years, hundreds of dollars) and the Lean verification mechanism.
- [Google решил 9 математических задач, над которыми ученые бились десятилетиями](https://hi-tech.mail.ru/news/157491-google-reshil-9-matematicheskih-zadach-nad-kotorymi-uchenye-bilis-desyatiletiyami/) - Confirms combining AlphaProof with LLMs for solution searching and proof verification.
- [Google DeepMind смогла решить ещё девять открытых задач Эрдёша и десятки математических гипотез](https://www.ixbt.com/news/2026/05/27/google-deepmind-smogla-reshit-eshjo-devjat-otkrytyh-zadach-jerdjosha-i-desjatki-matematicheskih-gipotez.html) - Previously reported (May 27, 2026) regarding the same AlphaProof Nexus framework and nine Erdős problems; numerical metrics are consistent.

## FAQ

### How many Erdős problems did AlphaProof Nexus solve?

The system solved 9 out of 353 open problems associated with Paul Erdős; two of them remained unsolved for 56 years.

### How does AlphaProof Nexus differ from a regular chatbot?

It uses Lean for formalization: the AI outputs a proof in formal format, and Lean checks every logical step and returns feedback upon error. A convincing text explanation is not enough.

### How expensive is it?

According to the authors, computing for each solved Erdős problem cost a few hundred dollars.

### What else can the system do?

It proved 44 out of 492 open OEIS conjectures, helped with an algebraic geometry problem, improved bounds in convex optimization, found formalization errors in literature, and is used in graph theory and quantum optics.