An exact formula for Erdős' problem 1005

Abstract

In 1943, Erdős considered the minimum number $f(n)$ of terms between two fractions in the Farey sequence of order $n$ whose numerators and denominators are oppositely ordered. Determining the constant $c$ in $f(n)=(c+o(1))n$ is known as Erdős Problem 1005. Recently, Cipollini solved this asymptotic problem by proving that $f(n)=(1/4+o(1))n$. Following his framework, we give an analytic proof of an exact formula for $f(n)$ for all sufficiently large $n$. Combining this with a finite computer verification, we further determine $f(n)$ for every integer $n\geq 4$.