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A VARIATION ON THE THEME OF NICOMACHUS

Part of: Finite fields and commutative rings (number-theoretic aspects) Arithmetic algebraic geometry Computational number theory

Published online by Cambridge University Press:  28 March 2018

FLORIAN LUCA Open the ORCID record for FLORIAN LUCA [Opens in a new window] ,
GEREMÍAS POLANCO  and
WADIM ZUDILIN Open the ORCID record for WADIM ZUDILIN [Opens in a new window]
FLORIAN LUCA
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Department of Mathematics, Faculty of Sciences, University of Ostrava, 30 dubna 22, 701 03 Ostrava 1, Czech Republic email Florian.Luca@wits.ac.za
GEREMÍAS POLANCO
Affiliation:
School of Natural Science, Hampshire College, 893 West St, Amherst, MA 01002, USA email gpeNS@hampshire.edu
WADIM ZUDILIN*
Affiliation:
IMAPP, Radboud Universiteit, PO Box 9010, 6500 GL Nijmegen, The Netherlands School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia email w.zudilin@math.ru.nl, wadim.zudilin@newcastle.edu.au
*
✉w.zudilin@math.ru.nl, wadim.zudilin@newcastle.edu.au
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Abstract

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In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.

Keywords

elliptic curvesgeneratorsfinite fields

MSC classification

Primary: 11G20: Curves over finite and local fields
Secondary: 11T23: Exponential sums 11Y16: Algorithms; complexity
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 367 - 373
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2018 Australian Mathematical Publishing Association Inc.

Footnotes

The first author was supported in part by NRF (South Africa) Grant CPRR160325161141 and an A-rated researcher award, and by CGA (Czech Republic) Grant 17-02804S. The second author was supported in part by the Institute of Mathematics of Universidad Autonoma de Santo Domingo, Grant FONDOCyT 2015-1D2-186, Ministerio de Educación Superior Ciencia y Tecnología (Dominican Republic).

References

Kimberling, C., ‘The Zeckendorf array equals the Wythoff array’, Fibonacci Quart. 33 (1995), 3–8.Google Scholar
Stopple, J., A Primer of Analytic Number Theory. From Pythagoras to Riemann (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Warnaar, S. O., ‘On the q-analogue of the sum of cubes’, Electron. J. Combin. 11(1) (2004), Note 13, 2 pages.Google Scholar