On the Index of Farey Sequences

For any two consecutive Farey fractions γi = ai/qi < γi+1 = ai+1/qi+1, one has ai+1qi − aiqi+1 = 1 and qi + qi+1 > Q. Conversely, if q and q ′ are two coprime integers in {1, . . . , Q} with q + q > Q, then there are unique a ∈ {1, . . . , q} and a ∈ {1, . . . , q} for which aq − aq = 1, and a/q < a/q are consecutive Farey fractions of order Q. Therefore, the pairs of coprime integers (q, q) with q + q > Q are in one-to-one correspondence with the pairs of consecutive Farey fractions of order Q. Moreover, the denominator qi+2 of γi+2 can be easily expressed (cf. [5]) by means of the denominators of γi and γi+1 as qi+2 = [