Length of continued fractions in principal quadratic fields

Let d ≥ 2 be a square-free integer and for all n ≥ 0, let ${\displaystyle l\left({(\surd d)}^{2n+1}\right)}$ be the length of the continued fraction expansion of ${\displaystyle {(\surd d)}^{2n+1}}$. 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 ${\displaystyle C₁{(\surd d)}^{2n+1}\ge l\left({(\surd d)}^{2n+1}\right)\ge C₂{(\surd d)}^{2n+1}}$ 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].