Human-assisted Proof Identifies {1, 2, 4, 8, 13} As Counterexample to $1000 Erdős Prize Conjecture on Sidon Subsets

A long-standing problem in number theory, repeatedly posed as a conjecture by Paul Erdős, concerns the relationship between Sidon sets and perfect difference sets, and mathematicians have sought to determine if every finite Sidon set can be extended to a finite perfect difference set. Boris Alexeev and Dustin G. Mixon, from The Ohio State University, now resolve this question, demonstrating that such an extension is not always possible. Their work establishes that the set {1, 2, 4, 8, 13} serves as a counterexample to Erdős’s conjecture, overturning a problem that remained open for decades. Interestingly, the team also uncovered a previously overlooked counterexample published by Marshall Hall, Jr. decades earlier, and to ensure the validity of both findings, they developed a formal, machine-verified proof using artificial intelligence.

his conjecture. published a different counterexample three decades before Erdős first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, the team used ChatGPT to generate Lean code for proofs of both Hall’s and their own counterexamples. This paper largely follows the style of a standard mathematics paper, and readers primarily interested in the role of artificial intelligence within the work are advised to proceed directly to section 7, which details this aspect.

Formalizing Erdos Problem 707 with LLMs

This paper details the authors’ experience using a Large Language Model (LLM), specifically ChatGPT 5, to assist in formalizing Erdos Problem #707 within the Lean theorem prover. It presents a fascinating exploration of how LLMs can be integrated into the formal verification process, highlighting both the potential benefits and current limitations. The authors adopted a just-do-it proof style, focusing on directly implementing the mathematical logic in Lean rather than meticulously planning every step beforehand. This approach draws inspiration from the just-do-it proofs concept and the recent vibe coding technique, where code is generated based on intuition and a general understanding of the problem.

The team found ChatGPT 5 surprisingly effective at translating mathematical ideas into Lean code when combined with this iterative, vibe coding style, generating code snippets, suggesting definitions, and navigating the Lean syntax. The paper focuses on formalizing a specific problem from the long-standing list of unsolved problems posed by Paul Erdos: determining whether the Mian-Chowla sequence contains arbitrarily long arithmetic progressions. The team successfully formalized the problem statement, necessary definitions, and a significant portion of the required lemmas within the Lean library, providing a solid foundation for future work. ChatGPT 5 excelled at generating Lean code snippets for definitions and basic lemmas, proposing appropriate definitions based on mathematical descriptions, assisting with the often-complex Lean syntax and type system, and completing existing code.

However, the LLM lacks true understanding of the mathematics, operating instead based on pattern matching and statistical probabilities, which can lead to errors requiring careful review. The authors highlight the potential of integrating LLMs into the broader formal verification ecosystem, particularly with tools like the Lean mathematical library (mathlib) and the Formal Conjectures project. They emphasize the importance of creating a collaborative environment where humans and LLMs can work together to tackle complex mathematical problems. The paper suggests several avenues for future research, including developing more robust LLM-assisted formalization tools, exploring different prompting strategies to improve LLM performance, investigating the use of LLMs for automated theorem proving, and building a more comprehensive knowledge base of formalized mathematics. In essence, the paper presents a compelling case for the potential of LLMs as assistive tools in formal verification, accelerating the formalization process and empowering mathematicians and computer scientists to tackle previously intractable challenges.

Mian-Chowla Sequence Disproves Erdős Conjecture

Scientists have resolved a longstanding problem posed by Paul Erdős, a challenge that remained open for over half a century. The research definitively disproves a conjecture stating that every finite Sidon set can be extended to a finite perfect difference set. The team demonstrated this by identifying specific Sidon sets that cannot be so extended, providing concrete counterexamples to Erdős’s claim. Experiments revealed that the set {1, 2, 4, 8}, the first four terms of the Mian-Chowla sequence, does not extend to a finite perfect difference set modulo v = p2 + p + 1 for any prime p. had previously disproved the conjecture in 1947, a result that appears to have been overlooked for decades. Hall identified the set {-8, -6, 0, 1, 4} as a counterexample, which translates to {1, 3, 9, 10, 13} when shifted to positive integers. To ensure the rigor of their findings, the researchers developed a formal proof using the Lean proof assistant, with the majority of the code generated by ChatGPT. The resulting proof, spanning thousands of lines, formally verifies both Hall’s counterexample and the team’s own discoveries. This work not only resolves a significant mathematical problem but also demonstrates the potential of artificial intelligence tools in formal verification and mathematical discovery.

Lean Proves Erdős’s Sidon Set Conjecture False

This research resolves a longstanding problem proposed by Paul Erdős, concerning the extension of Sidon sets to perfect difference sets. The team definitively demonstrates that the set {1, 2, 4, 8, 13} serves as a counterexample to Erdős’s conjecture, disproving a problem that remained open for decades. To ensure the validity of both their own findings and Hall’s, the researchers constructed a formal, machine-verified proof using Lean, a sophisticated theorem prover. The resulting proof, exceeding 6000 lines of code, establishes the counterexamples with mathematical certainty. This work highlights the potential of formal verification tools and demonstrates the power of combining human expertise with automated reasoning to resolve complex mathematical challenges.