Lower bounds for Z-numbers

Dubickas, Artūras; Mossinghoff, Michael

doi:10.1090/S0025-5718-09-02211-X

Lower bounds for Z-numbers
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by Artūras Dubickas and Michael J. Mossinghoff;

Math. Comp. 78 (2009), 1837-1851

DOI: https://doi.org/10.1090/S0025-5718-09-02211-X

Published electronically: January 23, 2009

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Abstract:

Let $p/q$ be a rational noninteger number with $p>q\geq 2$. A real number $\lambda >0$ is a $Z_{p/q}$-number if $\{\lambda (p/q)^n\}<1/q$ for every nonnegative integer $n$, where $\{x\}$ denotes the fractional part of $x$. We develop several algorithms to search for $Z_{p/q}$-numbers, and use them to determine lower bounds on such numbers for several $p$ and $q$. It is shown, for instance, that if there is a $Z_{3/2}$-number, then it is greater than $2^{57}$. We also explore some connections between these problems and some questions regarding iterated maps on integers.

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