On the \(m\)-ary expansion of a multiple of \((m^k-1)/(m-1)\) (Q2875439)
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scientific article; zbMATH DE number 6330556
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the \(m\)-ary expansion of a multiple of \((m^k-1)/(m-1)\) |
scientific article; zbMATH DE number 6330556 |
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14 August 2014
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\(m\)-ary expansion
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sum of digits
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0.86334527
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0.8589811
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0.85865384
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0.85379416
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0.85287416
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0.8505893
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On the \(m\)-ary expansion of a multiple of \((m^k-1)/(m-1)\) (English)
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Let \(m\geq 2,~n\geq 1,~k\geq 1\) be integers. A conjecture of \textit{Z.-W Sun} [J. Aust. Math. Soc. 93, No. 1-2, 189--201 (2012; Zbl 1336.11020)] asserted that there are at least \(k\) nonzero digits in the \(m\)-ary expansion of \(n(m^k-1)/(m-1)\), and that the sum of the \(m\)-ary digits of \(n(m^k-1)\) is at least \(k(m-1)\). In this note, the author gives a proof of a stronger result which in particular implies Sun's conjecture.NEWLINENEWLINE Reviewer's note: The reviewer had also confirmed Sun's conjecture which he learned from Professor Andrzej Schinzel who attributed the conjecture to Bednarek (see \textit{F. Luca} [Commun. Math. 20, No. 2, 79--80 (2012; Zbl 1283.11016)]).
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