Multiplicative dependence of two integers shifted by a root of unity
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- by Paulius Drungilas and Artūras Dubickas PDF
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Abstract:
In this note we prove a result on the multiplicative independence of the numbers $m-\alpha ,n-\alpha$, where $m>n$ are positive integers and $\alpha$ is a reciprocal algebraic number with the property that $\alpha +1/\alpha$ has at least two real conjugates over $\mathbb {Q}$ lying in the interval $(-\infty ,2]$. As an application, we show that for any positive integers $m>n$ and $k \geqslant 3$ the numbers $m-\zeta _k, n-\zeta _k$, where $\zeta _k$ is the primitive $k$th root of unity, are multiplicatively independent except when $(n,k)=(1,6)$. This settles a recent conjecture of Madritsch and Ziegler.References
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Additional Information
- Paulius Drungilas
- Affiliation: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
- MR Author ID: 724133
- Email: pdrungilas@gmail.com
- Artūras Dubickas
- Affiliation: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
- Email: arturas.dubickas@mif.vu.lt
- Received by editor(s): May 2, 2017
- Received by editor(s) in revised form: February 21, 2018
- Published electronically: October 31, 2018
- Additional Notes: This research was funded by a grant (No. S-MIP-17-66/LSS-110000-1274) from the Research Council of Lithuania.
- Communicated by: Matthew Papanikolas
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 505-511
- MSC (2010): Primary 11R18; Secondary 11R04, 11A63, 11D41
- DOI: https://doi.org/10.1090/proc/14136
- MathSciNet review: 3894890