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adjacent fractions adjacents to h/k algebraic number angle application approximation automorphic function belong chapter circle of radius clearly common multiple complex numbers consider continued fraction convergent coprime decimal fraction decomposed denominator divided divisible divisor elementary elements equal example fact Farey sequence Figure finite group finite number fixed points Ford circles formula fundamental region geometry h/k and H/K Hence hinges hjki inequality infinity integers interior intersection interval inversion with respect irrational number isometric circle linear substitutions linear transformations linkage mediant mesh modular group modular substitution modular transformation norm pair parallelogram Pell's equation pigeon-hole principle point of tangency prime factors prime numbers problem proof prove rational numbers real axis reduced fraction remainder residues rhomboid sequence of order solutions straight line sub-group Suppose Sylvester's formula tangent theorem triangles twin primes unit circle upper half plane zero