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Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

Tony Feng, Trieu Trinh, Garrett Bingham, Jiwon Kang, Shengtong Zhang, Sang-hyun Kim, Kevin Barreto, Carl Schildkraut, Junehyuk Jung, Jaehyeon Seo, Carlo Pagano, Yuri Chervonyi, Dawsen Hwang, Kaiying Hou, Sergei Gukov, Cheng-Chiang Tsai, Hyunwoo Choi, Youngbeom Jin, Wei-Yuan Li, Hao-An Wu, Ruey-An Shiu, Yu-Sheng Shih, Quoc V. Le, Thang Luong

TL;DR

This paper presents a case study of semi-autonomous mathematics discovery using a Gemini-based AI agent (Aletheia) to systematically evaluate Open Erdős Problems. An AI natural-language verifier narrows the workload from 700 Open prompts to 212 promising candidates, which human experts then vet, leading to 63 technically correct and 13 meaningfully correct results across 200 AI responses. The study distinguishes fully autonomous solutions, partial AI contributions, independent rediscoveries, and literature identifications, and it reports five autonomous solutions alongside eight literature-based identifications, while highlighting risks such as notional misinterpretation and subconscious plagiarism. The authors advocate cautious optimism for AI-assisted math, emphasizing the need for robust literature-tracing, formal verification, and careful articulation of statements to avoid misinformation and overclaiming as AI approaches scale.

Abstract

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

TL;DR

This paper presents a case study of semi-autonomous mathematics discovery using a Gemini-based AI agent (Aletheia) to systematically evaluate Open Erdős Problems. An AI natural-language verifier narrows the workload from 700 Open prompts to 212 promising candidates, which human experts then vet, leading to 63 technically correct and 13 meaningfully correct results across 200 AI responses. The study distinguishes fully autonomous solutions, partial AI contributions, independent rediscoveries, and literature identifications, and it reports five autonomous solutions alongside eight literature-based identifications, while highlighting risks such as notional misinterpretation and subconscious plagiarism. The authors advocate cautious optimism for AI-assisted math, emphasizing the need for robust literature-tracing, formal verification, and careful articulation of statements to avoid misinformation and overclaiming as AI approaches scale.

Abstract

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.
Paper Structure (28 sections, 22 theorems, 127 equations, 1 figure, 3 tables)

This paper contains 28 sections, 22 theorems, 127 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. A semi-autonomous effort based on Gemini Deep Think
  3. Results
  4. The writing of this paper
  5. Contextualizing the results
  6. Discussion of other AI results on Erdős problems
  7. Conclusions
  8. Problems autonomously solved by AI
  9. Erdős-652
  10. Erdős-1051
  11. Problems with parts solved by AI
  12. Erdős-654
  13. Case 1: $u \in P$.
  14. Case 2: $u \in Q$.
  15. Erdős-935
  16. ...and 13 more sections

Key Result

Theorem 1

(Pach--Sharir PS98) Let $C$ be a set of $n$ simple curves in the plane with the property that For any set $P$ of $m$ points on the plane, the number of incidences between the points of $P$ and the curves of $C$ is bounded above by where $c(k,s)$ is a positive constant that depends on $k$ and $s$ but not $m$ or $n$. An incidence is a pair $(p,\gamma)$ with $p\in P$ and $\gamma\in C$.

Figures (1)

  • Figure :

Theorems & Definitions (57)

  • Remark 1.1
  • Remark 2.1
  • Theorem 1
  • Remark 2.2
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 47 more