Farey sequence and Graham's conjectures

[摘要] Let Fn be the Farey sequence of order n. For S subset of Fn we let Q(S) = {x/y : x, y E S, x < y and y not equal 0}. We show that if Q(S) C Fn, then vertical bar S vertical bar <= n + 1. Moreover, we prove that in any of the following cases: (1) Q(S) = F-n; (2) Q(S) C Fn and |S| = n + 1, we must have S = {0,1, 1/2, .. . , 1/n} or S = {0, 1, 1/n, ..., n-1/n } except for n = 4, where we have an n n , ... , n-1 n additional set {0, 1, 1/2, 1/3, 2/3} for the second case. Our results are based on Graham's GCD conjectures, which have been proved by Balasubramanian and Soundararajan. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Farey sequences;Graham's conjectures;Greatest common divisors