Problem Statement
Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey fractions of order $n\geq 4$. Let $f(n)$ be the largest integer such that if $1\leq k<l\leq k+f(n)$ then $\frac{a_k}{b_k}$ and $\frac{a_l}{b_l}$ are similarly ordered - in other words,\[(a_k-a_l)(b_k-b_l)\geq 0.\]Estimate $f(n)$ - in particular, is there a constant $c>0$ such that $f(n)=(c+o(1))n$ for all large $n$?
Categories:
Number Theory
Progress
The function $f(n)$ was first considered by Mayer [Ma42], who proved $f(n)\to \infty$ as $n\to \infty$. Erdős [Er43] proved $f(n)\gg n$.van Doorn [vD25b] has proved that\[\left(\frac{1}{12}-o(1)\right)n\leq f(n) \leq \frac{1}{4}n+O(1),\]and conjectures that the upper bound is optimal.
Source: erdosproblems.com/1005 | Last verified: January 19, 2026