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Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof.

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Title: Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof.
Authors: Alexeeva, Boris boris.alexeev@gmail.com, Mixon, Dustin G.1 mixon.23@osu.edu
Source: Proceedings of the National Academy of Sciences of the United States of America. 5/26/2026, Vol. 123 Issue 21, p1-9. 9p.
Subjects: Difference sets, Formal verification, Language models, Refutation (Logic), ChatGPT
Abstract: We resolve a $1,000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his "favorite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we found that although this problem was presumed to be open for half a century, Marshall Hall, Jr. 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, we used ChatGPT to vibe prove both Hall’s and our counterexamples in Lean. [ABSTRACT FROM AUTHOR]
Copyright of Proceedings of the National Academy of Sciences of the United States of America is the property of National Academy of Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof.
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  Data: <searchLink fieldCode="AR" term="%22Alexeeva%2C+Boris%22">Alexeeva, Boris</searchLink><i> boris.alexeev@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Mixon%2C+Dustin+G%2E%22">Mixon, Dustin G.</searchLink><relatesTo>1</relatesTo><i> mixon.23@osu.edu</i>
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  Data: We resolve a $1,000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his "favorite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we found that although this problem was presumed to be open for half a century, Marshall Hall, Jr. 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, we used ChatGPT to vibe prove both Hall’s and our counterexamples in Lean. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Proceedings of the National Academy of Sciences of the United States of America is the property of National Academy of Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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