Farey sequences have some very elegant properties, that even turn out to be useful. All references to terms of Farey sequences here refer to the reduced form of the fractions.

  • The difference between adjacent terms of a Farey sequence with denominators m and n is precisely 1/mn.
  • Suppose a/b and c/d are adjacent terms of the Farey sequence Fn (naturally, a,b < n). Then the two terms remain adjacent in all Farey sequences up to Fb+d, at which point (a+c) / (b+d) appears between them (and in particular, (a+c) / (b+d) is in reduced form!)
  • Let 0<x<1 be a real number. The convergents of the continued fraction for x are precisely the terms of Farey sequences Fn closest to x, in the order in which new terms appear with increasing n.

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