Ducci on $\mathbb{Z}_m^n$ and the maximum length for $n$ odd

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematical Sciences, Kent State University, Kent, OH, United States

Abstract

Let the Ducci function, $D$, be an endomorphism on $\mathbb{Z}_m^n$ such that \[D(x_1, x_2,\ldots,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m,\ldots, x_n+x_1 \; \text{mod} \; m).\] The sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Because $\mathbb{Z}_m^n$ is finite, the Ducci sequence of $\mathbf{u}$ enters a cycle for all $\mathbf{u} \in \mathbb{Z}_m^n$, which we call the Ducci cycle of $\mathbf{u}$. In this paper, our main goal is to prove that if $n$ is odd and $m=2^lm_1$ where $m_1$ is odd, then the longest it will take for a Ducci sequence in $\mathbb{Z}_m^n$ to enter its cycle is $l$ iterations of $D$. In addition to this, we will prove that the set of all tuples in $\mathbb{Z}_m^n$ in a Ducci cycle for some $\mathbf{u} \in \mathbb{Z}_m^n$ is $\{(x_1, x_2,\ldots,x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.

Keywords

Main Subjects

References

[1] A. T. Benjamin and J. J. Quinn, Proofs that really count. The art of combinatorial proof, The Dolciani Mathematical Expositions, 27, Mathematical Association of America, Washington, DC, 2003.
[2] F. Breuer,A note on a paper by H. Glaser and G. Sch¨offl, Fibonacci Quart., 36 no. 5 (1998) 463–466.
[3] F. Breuer, Ducci sequences over abelian groups, Comm. Algebra, 27 no. 12 (1999) 5999–6013.
[4] F. Breuer, Ducci sequences and cyclotomic fields, J. Difference Equ. Appl., 16 no. 7 (2010) 847–862.
[5] R. Brown and J. L. Merzel, The length of Ducci’s four-number game, Rocky Mountain J. Math., 37 no. 1 (2007) 45–65.
[6] P. Cameron, Combinatorics: topics, techniques, algorithms, Cambridge University Press, Cambridge, 1994.
[7] M. Chamberland, Unbounded Ducci sequences, J. Difference Equ. Appl., 9 no. 10 (2003) 887–895.
[8] B. Dular, Cycles of sums of integers, Fibonacci Quart., 58 no. 2 (2020) 126–139.
[9] A. Ehrlich, Periods in Ducci’s n-number game of differences, Fibonacci Quart., 28 no. 4 (1990) 302–305.
[10] B. Freedman, The four number game, Scripta Mathematica, 14 (1948) 35–47.
[11] H. Glaser and G. Sch¨offl, Ducci-sequences and Pascal’s triangle, Fibonacci Quart., 33 no. 4 (1995) 313–324.
[12] M. L. Lewis and S. M. Tefft, The Period of Ducci Cycles on Z2l for Tuples of Length 2k, Submitted for Publication, , 2024.
[13] A. L. Furno, Cycles of differences of integers, J. Number Theory, 13 no. 2 (1981) 255–261.
[14] M. Misiurewicz and A. Schinzel, On n numbers on a circle, Hardy-Ramanujan J., 11 (1988) 30–39.
[15] F. B. Wong, Ducci processes, Fibonacci Quart., 20 no. 2 (1982) 97–105.
International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 20 May 2025

Files

History

  • Receive Date: 19 September 2024
  • Revise Date: 13 May 2025
  • Accept Date: 18 May 2025
  • Published Online: 20 May 2025

Share

How to cite

Statistics