Convergence, finiteness and periodicity of several new algorithms of \(p\)-adic continued fractions (Q6590634)

Let \(p\) be a prime number, \(v_p(\cdot)\) the \(p\)-adic valuation and \(|\cdot|_p\) the \(p\)-adic norm. Let \(\mathcal{R}\subset \mathbb{Q}\) be a set of representatives modulo \(p\) such that \(0 \in \mathcal{R}\). It is well known that every \(\alpha \in \mathbb{Q}_p\) can written in the form\N\[\N\alpha=\sum_{n=r}^{\infty} a_n p^n,\N\]\Nwhere all \(a_n \in \mathcal{R}, r=v_p(\alpha)\) and \(a_r \neq 0\) if \(\alpha \neq 0\). Let \([b_0, b_1, b_2, \dots]\) be a continued fraction expansion of the number \(\alpha\). For a given \(\alpha \in \mathbb{Q}_p\) a sequence \(\{b_n \}\) is defined as the partial quotients of the \(p\)-adic continued fraction expansion. The authors present \(p\)-adic algorithms working with algebraic numbers that attempt to produce periodic \(p\)-adic continued fraction expansions for quadratic irrationals, along with finite expansions for rational numbers. The proposed algorithms always produce continued fractions that converge to a \(p\)-adic number. If \(\alpha =\frac{P}{Q} \in \mathbb{Q}\), then algorithms stop in ln \(\lceil\ln Q\rceil + 2 \) steps. For \(p=2,3\), the algorithms continually expand \(\alpha\) as an ultimately periodic continued fraction. The authors affirm that they propose a potential approach to establish \(p\)-adic continued fraction expansion algorithms for more significant primes \(p\). Some computational results are also presented in the paper.