By Lagrange’s theorem, the continued fraction expansion of a real positive irrational number is eventually periodic. It is still not known if a \(p\)-adic continued fraction algorithm exists that shares a similar property since a \(p\)-adic analogue of Lagrange’s theorem has not been proved yet. Some modifications and improvements to one of Browkin’s algorithms are investigated in the paper. This adjustment improves significantly the periodicity properties of the second Browkin’s algorithm. Let \(|\cdot|_p\) be the \(p\)-adic absolute value over \(\mathbb{Q}\), where \(p\) is an odd prime. A continued fraction of a value \(\alpha\) notated as \[ \alpha=[b_0,b_1,b_2,\ldots]. \] Given \(\alpha=\sum\limits_{i=-r}^\infty a_i p^i \in \mathbb{Q}_p,\) with \(a_i \in \big\{-\frac{p-1}{2}, \dots, \frac{p-1}{2} \big\}\), defined two floor functions as \[ s(\alpha)=\sum_{i=-r}^0 a_i p^i, \quad t(\alpha)=\sum_{i=-r}^{-1} a_i p^i, \] with \(r \in \mathbb{N}\), where \(s(\alpha)=0\) for \(r<0\) and \(t(\alpha)=0\) for \(r\leq 0\). The second Browkin’s algorithm is rewritten as \[ \begin{cases} b_n =s(\alpha_n) & \text{if \(n\) even} \\ b_n =t(\alpha_n) & \text{if \(n\) odd} \\ \alpha_{n+1}=\frac{1}{\alpha_n-b_n}. \end{cases} \] for all \(n \geq 0\) and given \(\alpha_0 \in \mathbb{Q}_p.\) A new algorithm shows better properties of periodicity.
Theorem 1. If \(\alpha_0 \in \mathbb{Q} \), then algorithm stops in a finite number of steps.
Theorem 2. If \(\alpha \in \mathbb{Q}_p\) has a periodic continued fraction expansion by means of algorithm, then the expansion is purely periodic if and only if \(|\alpha|_p\geq 1, \; |\bar{\alpha}| <1.\)
The authors provide examples of this algorithm.