Taiwanese Journal of Mathematics

THE RESTRICTED SOLUTIONS OF $ax+by=\gcd(a,b)$

Ju-Si Lee

Full-text: Open access

Abstract

Let $D$ denote a principle ideal domain with identity element $1$. Fix three elements $a, b, d$ in $D$ with ${\rm gcd}(a, b)=d$. We show there exist two elements $x, y$ in $D$ with either ${\rm gcd}(a, y)=1$ or ${\rm gcd}(b, x)=1$ such that $ax+by=d$. Moreover we show there exist $x, y$ in $D$ such that ${\rm gcd}(a, y)=1,$ ${\rm gcd}(b, x)=1$ and $ax+by=d$ if and only if for each prime divisor $p$ of $d$ with a complete set of residues modulo $p$ containing exactly $2$ elements, the power of $p$ appearing in the factorization of $a$ is different to that of $b$. We apply our results to the study of double-loop networks..

Article information

Source
Taiwanese J. Math., Volume 12, Number 5 (2008), 1191-1199.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500574257

Digital Object Identifier
doi:10.11650/twjm/1500574257

Mathematical Reviews number (MathSciNet)
MR2431889

Zentralblatt MATH identifier
1195.11008

Subjects
Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors

Keywords
principal ideal domain double-loop network

Citation

Lee, Ju-Si. THE RESTRICTED SOLUTIONS OF $ax+by=\gcd(a,b)$. Taiwanese J. Math. 12 (2008), no. 5, 1191--1199. doi:10.11650/twjm/1500574257. https://projecteuclid.org/euclid.twjm/1500574257


Export citation

References

  • D. S. Malik, John N. Mordeson and M. K. Sen, Fundamentals of Abstract Algebra, The McGraw-Hill Companies, Inc. 1997, Chapter 15.
  • H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J., 14 (1967), 1-27.
  • C. K. Wong and D. Coppersmith, A combinatorial Problem Related to Multimodule Organizations, J. ACM, 21 (1974), 392-402.
  • M. A. Fiol, M. Valero, J. L. A. Yebra and T. Lang, Optimization of Double-Loop equivalent Structures for Local Networks, Proc. 19th int'l Symp. MIMI'82, 1982, pp. 37-41.
  • F. K. Hwang and Y. H. Xu, Ddouble-loop networks with minimum delay, Discr Math., 66 (1987), 109-118.
  • F. K. Hwang and W.-C. W. Li, Reliabilities of double-loop networks, Prob Eng Inf Sci., 5 (1991), 255-272.