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 $n\geq 4$.

Disclosure

“al results in this paper, including (2) and the proof of Theorem 1.3, is available in a public GitHub repository at https://github.com/dct-cell/erdos/tree/main/1005. AI-use declaration During the preparation of this work, the authors used ChatGPT-5.6 to assist in generating candidate proof strategies. All AI-generated suggestions were verified and refined by the authors, who take full responsibility for the final content of the paper. References [1] A. E. Mayer, A mean value the”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:17:44.390124Z

Structural counts