# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## No two non-real conjugates of a Pisot number have the same imaginary partHTML articles powered by AMS MathViewer

by Artūras Dubickas, Kevin G. Hare and Jonas Jankauskas
Math. Comp. 86 (2017), 935-950 Request permission

## Abstract:

We show that the number $\alpha =(1+\sqrt {3+2\sqrt {5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $\alpha _1,\alpha _2,\alpha _3,\alpha _4$ satisfy the additive relation $\alpha _1+\alpha _2=\alpha _3+\alpha _4$. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations $\alpha _1 = \alpha _2 + \alpha _3+\alpha _4$ or $\alpha _1 + \alpha _2 + \alpha _3 + \alpha _4 =0$ cannot be solved in conjugates of a Pisot number $\alpha$. We also show that the roots of the Siegel’s polynomial $x^3-x-1$ are the only solutions to the three term equation $\alpha _1+\alpha _2+\alpha _3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $\alpha _1=\alpha _2+\alpha _3$.
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• Retrieve articles in Mathematics of Computation with MSC (2010): 11R06, 11R09
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• Artūras Dubickas
• Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
• Email: arturas.dubickas@mif.vu.lt
• Kevin G. Hare
• Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
• MR Author ID: 690847
• Email: kghare@uwaterloo.ca
• Jonas Jankauskas
• Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania – and – Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
• MR Author ID: 825362
• ORCID: 0000-0001-9770-7632
• Email: jonas.jankauskas@gmail.com
• Received by editor(s): May 12, 2015
• Received by editor(s) in revised form: August 21, 2015
• Published electronically: April 13, 2016
• Additional Notes: The research of the first and third authors was supported in part by the Research Council of Lithuania Grant MIP-068/2013/LSS-110000-740
The research of the second author was supported by NSERC Grant RGPIN-2014-03154.
Computational support was provided in part by the Canadian Foundation for Innovation, and the Ontario Research Fund.
• © Copyright 2016 American Mathematical Society
• Journal: Math. Comp. 86 (2017), 935-950
• MSC (2010): Primary 11R06; Secondary 11R09
• DOI: https://doi.org/10.1090/mcom/3103
• MathSciNet review: 3584555