Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof
Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proofWe resolve a $1000 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 “favourite” 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 discovered 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 code a Lean proof of both Hall’s and our counterexamples.
Boris Alexeev, ChatGPT, Lean, and Dustin G. Mixon
Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof
Proceedings of the National Academy of the Sciences (PNAS) U.S.A. 123 (21) e2531760123 (2026).
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