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Related Experiment Video

Updated: May 24, 2026

Generating Strictly Controlled Stimuli for Figure Recognition Experiments

Generating Strictly Controlled Stimuli for Figure Recognition Experiments

Published on: March 18, 2019

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

Boris Alexeev¹, Dustin G Mixon^2,3

¹Independent Researcher, Athens, GA 30605.

Proceedings of the National Academy of Sciences of the United States of America

|May 22, 2026

Summary

A long-standing mathematical conjecture by Paul Erdős is disproven. Researchers found a counterexample set, {1, 2, 4, 8, 13}, resolving a problem in combinatorial number theory.

Keywords:

combinatorics formal proofs large language models

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Published on: November 15, 2013

Area of Science:

Combinatorial Number Theory
Discrete Mathematics
Set Theory

Background:

Paul Erdős posed a conjecture in 1976 regarding finite Sidon sets and perfect difference sets.
The conjecture remained open for decades, becoming a notable problem in combinatorial mathematics.

Purpose of the Study:

To resolve the Erdős conjecture on extending finite Sidon sets to finite perfect difference sets.
To provide a counterexample to this long-standing mathematical problem.

Main Methods:

Identification of a specific finite set {1, 2, 4, 8, 13} as a counterexample.
Formal verification of the counterexample using a large language model (ChatGPT) within the Lean theorem prover.
Discovery of a prior counterexample by Marshall Hall, Jr.

Main Results:

The Erdős conjecture is disproven.
The set {1, 2, 4, 8, 13} serves as a concrete counterexample.
Marshall Hall, Jr. had previously published a different counterexample, predating Erdős's conjecture.

Conclusions:

The problem of extending finite Sidon sets to finite perfect difference sets is definitively closed.
Large language models can assist in formal verification of mathematical proofs.
Historical mathematical literature may contain overlooked solutions to prominent problems.