Geodesic
Length of continued fractions in principal quadratic fields
Acta Arithmetica, Tome 85 (1998) no. 1, pp. 35-49

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Let d ≥ 2 be a square-free integer and for all n ≥ 0, let $l((√d)^{2n+1})$ be the length of the continued fraction expansion of $(√d)^{2n+1}$. 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 $C₁(√d)^{2n+1} ≥ l((√d)^{2n+1}) ≥ C₂(√d)^{2n+1}$ 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].
DOI : 10.4064/aa-85-1-35-49

Guillaume Grisel 1

1 
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Guillaume Grisel. Length of continued fractions in principal quadratic fields. Acta Arithmetica, Tome 85 (1998) no. 1, pp. 35-49. doi: 10.4064/aa-85-1-35-49

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