Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof
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.
Fields of Study & Subjects
Limits and Structures in Graph Theory
graph theory and CDMA systems
Complexity and Algorithms in Graphs
Publication details
- Journal / source
- Proceedings of the National Academy of Sciences
- Volume
- 123
- Issue
- 21
- Pages
- e2531760123–e2531760123
- Publication date
- 22 May 2026
- Language
- EN