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On the Fractional Parts of Roots of Positive Real Numbers

Melvyn B. Nathanson
The American Mathematical Monthly
Vol. 120, No. 5 (May 2013), pp. 409-429
DOI: 10.4169/amer.math.monthly.120.05.409
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.120.05.409
Page Count: 21
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On the Fractional Parts of Roots of Positive Real Numbers
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Abstract

Abstract Let [ϕ] denote the integer part and {ϕ}the fractional part of the real number ϕ. For ϕ > 1 and [ϕ1/n] ≠ 0, define Mϕ(n) = [1/{ϕ1/n}]. The arithmetic function Mϕ(n) is eventually increasing, and \documentclass{article} \pagestyle{empty}\begin{document} $\lim_{n\rightarrow \infty} M_{\theta}(n)/n = 1/\log \theta$ \end{document} . Moreover, Mϕ(n) is “linearly periodic” if and only if log ϕ is rational. Other results and problems concerning the function Mϕ(n) are discussed.

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