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arXiv · 2509.14599

$S$-units and period length of continued fractions of linear recursions

Veekesh Kumar·Vivek Singh·Johannes Sprang
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Abstract

Let $(A_n)_{n\in \mathbb{Z}}$ be a linear recurrence sequence with values in a real quadratic field. In this paper, we study the question whether the period length of the continued fraction of $A_n$ is bounded as $n$ varies. The case where $(A_n)_n$ is a linear recurrence of degree $1$ has previously been solved by Corvaja and Zannier. Their result settled a problem posed by Mend\`es France about the length of the periods of the continued fractions for $\alpha^n$.

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Veekesh Kumar, Vivek Singh, Johannes Sprang. 2025-09-18. $S$-units and period length of continued fractions of linear recursions. https://arxiv.org/abs/2509.14599

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