Taiwanese Journal of Mathematics

ON THE TRIPLET-DIVISIONS OF THE SET $\{{ 1,2,3,\cdots ,p-2}\}$

Chyi-Lung Lin

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Abstract

In this paper, we give a number theoretical application of a simple discrete map. For any prime $p\geq 5$, we show that the numbers in the set $G(1, p-2)=\{1, 2, 3, \cdots , p-2\}$ can be divided into disjoint triplets $\langle a, b, c\rangle $, such that $a\cdot b\cdot c\equiv 1$ (mod $p$). This is a generalization of the well known doublet--division, that is, the numbers in the set $\{1, 2, 3, \cdots , p-1\}$ can be divided into disjoint pairs $\langle a, a^*\rangle $, such that $a\cdot a^*\equiv 1$ (mod $p$). The reason that we can perform the triplet-division is because there exists a simple map $F$ defined on the set $G(1, p-2)$ such that any number $a\in G(1, p-2)$ is a period-3 point of the map. Furthermore, the triplet $\langle a,~F(a),~F^2(a)\rangle $ has the property that $a\cdot F(a)\cdot F^2(a)\equiv 1$ (mod $p$).

Article information

Source
Taiwanese J. Math., Volume 2, Number 3 (1998), 303-312.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406970

Digital Object Identifier
doi:10.11650/twjm/1500406970

Mathematical Reviews number (MathSciNet)
MR1641163

Zentralblatt MATH identifier
0913.11003

Subjects
Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 11A41: Primes 11B75: Other combinatorial number theory

Keywords
prime doublet-division triplet-division iterated maps

Citation

Lin, Chyi-Lung. ON THE TRIPLET-DIVISIONS OF THE SET $\{{ 1,2,3,\cdots ,p-2}\}$. Taiwanese J. Math. 2 (1998), no. 3, 303--312. doi:10.11650/twjm/1500406970. https://projecteuclid.org/euclid.twjm/1500406970


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