Theoretical Analysis of Farey Sequences of Order n and Their Mathematical Properties - Sadat, Sadia (Journal of Natural Sciences – Kabul University, 2025)
Farey sequences are sets of irreducible fractions between 0 and 1, arranged in ascending order, with denominators less than or equal to a given natural number n. Each Farey sequence begins with 0/1 and ends with 1/1 and contains only fractions less than or equal to one. This paper presents a theoretical and analytical review of Farey sequences, focusing on their structural properties and mathematical significance. The investigation highlights the wide-ranging applications of these sequences in number theory, discrete mathematics, combinatorics, cryptography, spatial statistics, and even physics and economics—emphasizing their interdisciplinary relevance. A key property examined is the relationship between consecutive terms, characterized by the determinant condition ad−bc=1ad - bc = 1ad−bc=1, which reflects the adjacency of neighboring fractions. The paper also explores the mediant property used to generate new fractions within the sequence. Additionally, it discusses the role of Farey sequences in providing optimal rational approximations to irrational numbers, demonstrating their utility in both pure and applied mathematics.