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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Lower bounds for Z-numbers
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by Artūras Dubickas and Michael J. Mossinghoff PDF
Math. Comp. 78 (2009), 1837-1851 Request permission

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.
References
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Additional Information
  • Artūras Dubickas
  • Affiliation: Department of Mathematics and Informatics, Vilnius University Naugarduko 24, LT-03225 Vilnius, Lithuania
  • Email: arturas.dubickas@mif.vu.lt
  • Michael J. Mossinghoff
  • Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996
  • MR Author ID: 630072
  • ORCID: 0000-0002-7983-5427
  • Email: mimossinghoff@davidson.edu
  • Received by editor(s): January 22, 2008
  • Received by editor(s) in revised form: August 7, 2008
  • Published electronically: January 23, 2009
  • Additional Notes: The research of the first author was partially supported by the Lithuanian State Science and Studies Foundation.
  • © Copyright 2009 American Mathematical Society
  • Journal: Math. Comp. 78 (2009), 1837-1851
  • MSC (2000): Primary 11K31; Secondary 11J71, 11Y35
  • DOI: https://doi.org/10.1090/S0025-5718-09-02211-X
  • MathSciNet review: 2501079