Hostname: page-component-54dcc4c588-54gsr Total loading time: 0 Render date: 2025-10-05T02:40:56.155Z Has data issue: false hasContentIssue false
  • English
  • Français

LOWER BOUNDS FOR PERIODS OF DUCCI SEQUENCES

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

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 31 - 38
Check for updates
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

Footnotes

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

References

Ahmadi, O., Shparlinski, I. E. and Voloch, J. F., ‘Multiplicative order of Gauss periods’, Internat. J. Number Theory 6 (2010), 877–882.10.1142/S1793042110003290CrossRefGoogle Scholar
Andrews, G. E., The Theory of Partitions (Addison-Wesley, New York, 1976).Google Scholar
Breuer, F., ‘Periods of Ducci sequences and odd solutions to a Pellian equation’, Bull. Aust. Math. Soc. 100 (2019), 201–205.CrossRefGoogle Scholar
Breuer, F., Lötter, E. and van der Merwe, A. B., ‘‘Ducci sequences and cyclotomic polynomials’, Finite Fields Appl. 13 (2007), 293–304.CrossRefGoogle Scholar
Brown, R. and Merzel, J. L., ‘The number of Ducci sequences with given period’, Fibonacci Quart. 45 (2007), 115–121.Google Scholar
Calkin, N. J., Stevens, J. G. and Thomas, D. M., ‘A characterization for the length of cycles of the n-number Ducci game’, Fibonacci Quart. 43 (2005), 53–59.Google Scholar
Ciamberlini, C. and Marengoni, A., ‘Su una interessante curiosità numerica’, Periodiche di Matematiche 17 (1937), 25–30.Google Scholar
Clausing, A., ‘Ducci matrices’, Amer. Math. Monthly 125 (2018), 901–921.CrossRefGoogle Scholar
Ehrlich, A., ‘Periods of Ducci’s N-number game of differences’, Fibonacci Quart. 28 (1990), 302–305.Google Scholar
Erdős, P. and Murty, M. R., ‘On the order of a (mod p)’, in: Proc. 5th Canadian Number Theory Association Conf. (American Mathematical Society, Providence, RI, 1999), 87–97.Google Scholar
von zur Gathen, J. and Shparlinski, I. E., ‘Orders of Gauss periods in finite fields’, Appl. Algebra Engrg. Comm. Comput. 9 (1998), 15–24.CrossRefGoogle Scholar
Glaser, H. and Schöffl, G., ‘Ducci-sequences and Pascal’s triangle’, Fibonacci Quart. 33 (1995), 313–324.Google Scholar
Hagis, P., ‘On a class of partitions with distinct summands’, Trans. Amer. Math. Soc. 112 (1964), 401–415.CrossRefGoogle Scholar
Kurlberg, P. and Pomerance, C., ‘On the period of the linear congruential and power generators’, Acta Arith. 119 (2005), 149–169.CrossRefGoogle Scholar
Ludington, A. L., ‘Cycles of differences of integers’, J. Number Theory 13 (1981), 255–261.Google Scholar
Misiurewicz, M., Stevens, J. G. and Thomas, D. M., ‘Iterations of linear maps over finite fields’, Linear Algebra Appl. 413 (2006), 218–234.CrossRefGoogle Scholar
Moree, P., ‘Artin’s primitive root conjecture – a survey’, Integers 12A (2012), 1–100; Paper A13.Google Scholar
The On-line Encyclopedia of Integer Sequences, entry $\#$A038553. https://oeis.org/A038553.Google Scholar
Popovych, R., ‘Elements of high order in finite fields of the form 𝔽q[x]/𝛷r(x)’, Finite Fields Appl. 18(4) (2012), 700–710.CrossRefGoogle Scholar
Popovych, R., ‘Sharpening of the explicit lower bounds for the order of elements in finite field extensions based on cyclotomic polynomials’, Ukrainian Math. J. 66(6) (2014), 916–927.CrossRefGoogle Scholar
Shparlinski, I. E., ‘Linear equations with rational fractions of bounded height and stochastic matrices’, Q. J. Math. 69 (2018), 487–499.CrossRefGoogle Scholar
Silverman, J. H., ‘Wieferich’s criterion and the abc-conjecture’, J. Number Theory 30 (1988), 226–237.CrossRefGoogle Scholar
Solak, S. and Bahşi, M., ‘Some properties of circulant matrices with Ducci sequences’, Linear Algebra Appl. 542 (2018), 557–568.CrossRefGoogle Scholar