Abstract
Let Fn be the Farey sequence of order n. For S⊆Fn we let Q(S)={x/y:x,y∈S,x≤y and y≠0}. We show that if Q(S)⊆Fn, then |S|≤n+1. Moreover, we prove that in any of the following cases: (1) Q(S)=Fn; (2) Q(S)⊆Fn and |S|=n+1, we must have S={0,1,12,…,1n} or S={0,1,1n,…,n−1n} except for n=4, where we have an additional set {0,1,12,13,23} for the second case. Our results are based on Graham's GCD conjectures, which have been proved by Balasubramanian and Soundararajan.
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