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Length of continued fractions in principal quadratic fields

Guillaume Grisel — 1998

Acta Arithmetica

Let d ≥ 2 be a square-free integer and for all n ≥ 0, let l ( ( d ) 2 n + 1 ) be the length of the continued fraction expansion of ( d ) 2 n + 1 . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that C ( d ) 2 n + 1 l ( ( d ) 2 n + 1 ) C ( d ) 2 n + 1 for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].

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