Positive Lower Density for Hofstadter's $ab-1$ Problem

Samuel Korsky

arXiv:2608.07910v1 Primary: math.NT Secondary: math.CO

Abstract

Let $A$ be the smallest set of positive integers containing $2$ and $3$ such that $ab-1\in A$ whenever $a,b\in A$ are distinct. We prove that $A$ has positive lower density, answering a problem of Erdős attributed to Hofstadter.

Disclosure

“≥ > 0. x 17q d Acknowledgments The author thanks Thomas Bloom for helpful advice on the exposition, and Boris Alexeev for formalizing the proof in Lean [1]. GPT-5.6 Pro assisted in searching for and checking the finite interval and drift data. The author verified the argument and takes responsibility for the proof. References [1] B. Alexeev and Codex, Lean formalization of a solution to Erdős Prob”
PDF page 7

Classification: Computational experiments or data processing
Multiplier: 3
Verified: 2026-08-23T05:01:06.107534Z

Structural counts

Pages 8 pdf

Theorems 1 source

Lemmas 2 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 42 source

Bibliography entries 20 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file Hofstadter.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.