## Integer Solutions of the Diophantine Equation $$\left (1-\frac{1}{\mathit{x}}\right)$$ $$\left (1-\frac{1}{\mathit{y}}\right)$$ $$\left (1-\frac{1}{\mathit{z}}\right)$$ = $$\frac{1}{\mathit{l}}$$

Published: 2023-11-07

Page: 115-122

Qiang Wang *

Huaide College, Changzhou University, P.R. China and Aliyun School of Big Data, Changzhou University, P.R. China.

Qingwei Tu

Huaide College, Changzhou University, P.R. China and Aliyun School of Big Data, Changzhou University, P.R. China.

*Author to whom correspondence should be addressed.

### Abstract

In this paper, we mainly find all solutions of the diophantine equation $$\left (1-\frac{1}{\mathit{x}}\right)$$ $$\left (1-\frac{1}{\mathit{y}}\right)$$ $$\left (1-\frac{1}{\mathit{z}}\right)$$ = $$\frac{1}{\mathit{l}}$$ in integer variables ($$\mathit{x}$$, $$\mathit{y}$$, $$\mathit{z}$$, $$\mathit{l}$$ ).

Keywords: Diophantine equation, integer solution

#### How to Cite

Wang, Q., & Tu, Q. (2023). Integer Solutions of the Diophantine Equation $$\left (1-\frac{1}{\mathit{x}}\right)$$ $$\left (1-\frac{1}{\mathit{y}}\right)$$ $$\left (1-\frac{1}{\mathit{z}}\right)$$ = $$\frac{1}{\mathit{l}}$$. Asian Research Journal of Mathematics, 19(11), 115–122. https://doi.org/10.9734/arjom/2023/v19i11758

### References

Zhao KE, Qi SUN. Lectures on the Number Theory[M]. Beijing:Higher Education Press(2nd); 2002.

Zhao KE, Qi SUN. About Indeterminate Equation[M]. Harbin:Harbin Institute of Technology; 2011.

Fuzhen CAO. Introduction to Diophantine Equations[M]. Harbin:Harbin Institute of Technology; 2012.

Yi WU, Zhengping ZHANG. The Positive Integer Solutions of a Diophantine Equation. International Conference on Mechatronics Engineering and Modern Technologies in Industrial Engineering; 2015.

V. O. Osipyan, K. I. Litvinov, etc. Development of information security system mathematical models by the solutions of the multigrade Diophantine equation systems. Proceedings of the 12th International Conference on Security of Information and Networks; 2019.

Georgios Feretzakis, Dimitris Kalles, etc. On Using Linear Diophantine Equations for Efficient Hiding of Decision Tree Rules. Proceedings of the 10th Hellenic Conference on Arti cial IntelligenceJuly; 2018.

Zhen BAO, Fang LI. A study on Diophantine equations via cluster theory[J]. Journal of Algebra; 2023.

Michael Monagan, Baris Tuncer. Using sparse interpolation to solve multivariate diophantine equations. ACM Communications in Computer Algebra; 2015.

MORDELL L J. Diophantine Equations[M]. Salt Lake City UT: Academic Press; 1969.