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PERIODS OF DUCCI SEQUENCES AND ODD SOLUTIONS TO A PELLIAN EQUATION

Part of: Sequences and sets Diophantine equations Algebraic number theory: global fields

Published online by Cambridge University Press:  03 July 2019

FLORIAN BREUER Open the ORCID record for FLORIAN BREUER [Opens in a new window]
FLORIAN BREUER*
Affiliation:
University of Newcastle, Callaghan, NSW 2308, Australia email [email protected]
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Abstract

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A Ducci sequence is a sequence of integer $n$-tuples generated by iterating the map

$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$
Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$. We prove a new upper bound in the case where $n$ is a power of a prime $p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ for which $2$ is a primitive root and the Pellian equation $x^{2}-py^{2}=-4$ has no solutions in odd integers $x$ and $y$.

Keywords

Ducci sequencePellian equationcyclotomic fieldreal quadratic unit

MSC classification

Primary: 11B83: Special sequences and polynomials
Secondary: 11D09: Quadratic and bilinear equations 11R18: Cyclotomic extensions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 201 - 205
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
© 2019 Australian Mathematical Publishing Association Inc.

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