Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof
Boris Alexeev, Dustin G. Mixon
TL;DR
The paper disproves Erdős’s conjecture that every finite Sidon set can be embedded into a finite perfect difference set by exhibiting counterexamples, notably $\{1,2,4,8\}$ for prime-modulus contexts and $\{1,2,4,8,13\}$ for all moduli. It links perfect difference sets to cyclic projective planes via Sidon-set modular representations, applying polarity arguments (following Hall) to derive contradictions. A key feature is a human-assisted Lean formalization of the arguments, with a ChatGPT-generated Lean proof that verifies the counterexamples and clarifies subtleties such as a hidden $v=0$ case in a formalization of Erdős problem 707. The work demonstrates a concrete, large-language-model–assisted approach to formal mathematics, addressing both the mathematics and the practicalities/limitations of AI in proving intricate combinatorial-design statements. It resolves a long-standing Erdős prize problem and outlines future directions for forbidden Sidon sets and AI-assisted mathematical research.
Abstract
We 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.