Erdős Problem #707: Sidon Sets and Perfect Difference Sets
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Statement
Erdős conjectured, in over a dozen papers spanning 1976 to 1997 and with a $1000 prize attached, that every finite Sidon set extends to a perfect difference set modulo for some prime . Alexeev and Mixon establish that is a counterexample - and discovered along the way that Marshall Hall, Jr. had published a different counterexample three decades before Erdős first posed the problem, unnoticed by the community for half a century.
Context
Hall's 1947 counterexample predates the problem itself; this paper's counterexample is independent, smaller, and Lean-certified.
A $1000 Erdős prize problem - among his highest bounties - posed in more than a dozen papers across two decades and described as one of his favourite conjectures.
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The mathematics is the humans'; the paper is candid that LLMs failed at the two things they are usually praised for here - they never located Hall's paywalled prior solution, and "even after we knew what exactly to prove, it couldn't help us close the gap." What ChatGPT did do: write the complete Lean formalization of both counterexamples ("we decided to vibe code the whole proof... about a week... somehow it succeeded"). Earlier versions of the paper listed ChatGPT and Lean as authors until arXiv policy required their removal.
Reviews
0 human reviews · 1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Lean formalization of the result
Both Hall's and the new counterexample are formalized and kernel-checked in Lean, with the formalization written by ChatGPT and audited by the authors; erdosproblems.com marks the problem disproved with the proof verified in Lean.
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