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LOWER BOUNDS FOR PERIODS OF DUCCI SEQUENCES

Part of: Additive number theory; partitions Sequences and sets Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  22 November 2019

FLORIAN BREUER Open the ORCID record for FLORIAN BREUER [Opens in a new window]  and
IGOR E. SHPARLINSKI Open the ORCID record for IGOR E. SHPARLINSKI [Opens in a new window]
FLORIAN BREUER
Affiliation:
School of Mathematical and Physical Sciences,University of Newcastle, Newcastle, NSW 2308, Australia email florian.breuer@newcastle.edu.au
IGOR E. SHPARLINSKI*
Affiliation:
School of Mathematics and Statistics,University of New South Wales, Sydney, NSW 2052, Australia email igor.shparlinski@unsw.edu.au
*
✉igor.shparlinski@unsw.edu.au
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Abstract

A Ducci sequence is a sequence of integer $n$-tuples obtained 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 odd $n$. We prove a lower bound for $P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo $n$.

Keywords

Ducci sequencefinite fieldmultiplicative orderpartition

MSC classification

Primary: 11B83: Special sequences and polynomials
Secondary: 11P83: Partitions; congruences and congruential restrictions 11T30: Structure theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 31 - 38
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was supported in part by the Australian Research Council Grant DP180100201.

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