Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE STRUCTURE OF CERTAIN SUBSET OF FAREY SEQUENCE

  • Xing-Wang Jiang (Department of Mathematics Luoyang Normal University) ;
  • Ya-Li Li (School of Mathematics and Statistics Henan University)
  • Received : 2022.06.09
  • Accepted : 2023.03.30
  • Published : 2023.07.31
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Abstract

Let Fn be the Farey sequence of order n. For S ⊆ Fn, let 𝓠(S) be the set of rational numbers x/y with x, y ∈ S, x ≤ y and y ≠ 0. Recently, Wang found all subsets S of Fn with |S| = n + 1 for which 𝓠(S) ⊆ Fn. Motivated by this work, we try to determine the structure of S ⊆ Fn such that |S| = n and 𝓠(S) ⊆ Fn. In this paper, we determine all sets S ⊆ Fn satisfying these conditions for n ∈ {p, 2p}, where p is prime.

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Acknowledgement

This work was supported by the National Natural Science Foundations of China, Grant Nos. 12171243, 11901156 and 12201281, and the Natural Science Foundation of Youth of Henan Province, Grant No. 222300420245.

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