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Abstract <p> Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(D \ge 2\)</EquationSource> </InlineEquation> be an integer, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Phi_Q\)</EquationSource> </InlineEquation> be a classical Farey sequence of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(Q\)</EquationSource> </InlineEquation>. Let us mark the fractions of this sequence whose denominators belong to a given arithmetic progression with difference <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(D\)</EquationSource> </InlineEquation>. What is the maximum number of fractions in the interval between two adjacent marked fractions? In this paper, we obtain precise formulas for these quantities in the case of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(D=3\)</EquationSource> </InlineEquation>. </p>

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On Some Arithmetic Properties of Farey Fractions

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • K. S. Arzhanykh

摘要

Abstract

Let \(D \ge 2\) be an integer, and let \(\Phi_Q\) be a classical Farey sequence of order \(Q\) . Let us mark the fractions of this sequence whose denominators belong to a given arithmetic progression with difference \(D\) . What is the maximum number of fractions in the interval between two adjacent marked fractions? In this paper, we obtain precise formulas for these quantities in the case of \(D=3\) .