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A characterization of rational numbers by \(p\)-adic Sylvester series expansions. (English) Zbl 1140.11003

It is proved that a \(p\)-adic number is rational if and only if its \(p\)-adic Sylvester series expansion, defined via the Knopfmachers’ algorithm, is either finite or infinite of a particular shape. Note that the situation for fields of formal Laurent series over a finite field is much simpler (see a paper by the same authors and A. Harnchoowong [Thai J. Math. 4, 223–244 (2006)]

MSC:

11A67 Other number representations
11J61 Approximation in non-Archimedean valuations
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