Convergence conditions for \(p\)-adic continued fractions (Q6095449)

Let \(\nu_p(\cdot)\) and \(|\cdot|_p\) be the \(p\)-adic valuation and the \(p\)-adic absolute value over \(\mathbb{Q}\), where \(p\) is an odd prime. A continued fraction of a value \(\alpha\) notated as \[ \alpha=[b_0,b_1,b_2,\ldots]. \] Let \(A_n/B_n, n \in \mathbb{N},\) be the convergents of the continued fraction. A two Browkin algorithms for expansion \(\alpha \in \mathbb{Q}_p\) into a \(p\)-adic continued fraction are well known. The some modification of second Browkin algorithm was propose in this paper. The following theorem is proven. Theorem. The following conditions are equivalent: \[ (\mathrm{i}) \; \nu_p(b_{n+1}B_n) < \nu_p(B_{n-1}), \; \text{for all} \; n \geq 1, \quad (\mathrm{ii}) \; \nu_p(b_{n}b_{n+1}) < 0, \; \text{for all} \; n \geq 1. \] A new algorithms designed authors. The constructed continued fraction \([b_0, b_1,\ldots]\) is convergent to a \(p\)-adic number.