Optimality of Wouter van Doorn's Upper Bound for the Mayer-Erdős Farey Problem

Abstract

Let $\mathcal{F}_n$ be the Farey sequence of order $n$, written in increasing order. Call two fractions $\frac{a}{b} < \frac{c}{d}$ badly ordered if $a < c$ and $b > d$. Let $f(n)$ be the minimum number of Farey fractions strictly between two badly ordered fractions in $\mathcal{F}_n$. We prove $f(n)=\left(\frac{1}{4}+o(1)\right)n$. In the equivalent indexing convention of Erdős Problem 1005, this determines the requested asymptotic constant as $c=1/4$. The upper bound $f(n)\le n/4+O(1)$ was first obtained by Wouter van Doorn; the main result here is the matching lower bound.

Disclosure

“Acknowledgments and contribution statement This paper was written by GPT-5.5 Pro. The mathematical findings and proof strategy are due to Ricky Cipollini together with GPT-5.5 Thinking. The upper bound f (n) ≤ n/4 + O(1) and the conjecture that this is the optimal constant are due to Wouter van Doorn [3]; that cont”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:06:17.769129Z

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