Note on \(p\)-adic continued fractions
This paper is concerned with p-adic continued fraction expansions (in the sense of Mahler, as modified by Browkin). In analogy with the classical theorem of Galois the author shows: if the \(p\)-adic number \(\xi\) has a periodic \(p\)-adic continued fraction expansion (which implies that it is a quadratic number), then the expansion is purely periodic if and only if \(\|\xi\|_p > 1\) and \(\|\bar\xi\|^p < 1\) (\(\bar\xi\) denotes the conjugate of \(\xi\)). He also shows: if \(m \in Z\) and \(\sqrt{m}\) is a \(p\)-adic number, having a \(p\)-adic periodic continued fraction expansion, then the pre-period has length 3 when \(p = 2\) and \(m \equiv 4 \pmod 8\), otherwise it has length 2. The paper concludes with tables illustrating the periodicity of \(p\)-adic continued fractions expansions of various numbers \(\sqrt{m}\).
scientific article; zbMATH DE number 4083774