2026/07/25 by Ricky Cipollini
#math.NT
Let Fn be the Farey sequence of order n, written in increasing order. Call two fractions (a)/(b) < (c)/(d) badly ordered if a < c and b > d. Let f(n) be the minimum number of Farey fractions strictly between two badly ordered fractions in Fn. We prove f(n)=((1)/(4)+o(1))n. In the equivalent indexing convention of Erdős Problem 1005, this determines the requested asymptotic constant as c=1/4. The upper bound f(n)≤ n/4+O(1) was first obtained by Wouter van Doorn; the main result here is the matching lower bound.