.svg){kind=logo}
A DIOPHANTINE EQUATION AND ITS POSITIVE INTEGER SOLUTIONS
Keywords:
Diophantine equation, positive integer solution, greatest common divisor, prime numberDOI:
https://doi.org/10.17654/0972555522024Abstract
In 2017 , the Diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3 p}$ (*) was preliminarily discussed [12]. The positive integer solutions of some equations derived from $(*)$ were solved, where $p=661$. In this paper, this equation $(*)$ is further discussed and its results are supplemented. All the positive integer solutions of the remaining unsolved equations are solved by using simple and elementary methods, which are verified by computer calculation.
Received: April 17, 2022
Revised: May 10, 2022
Accepted: June 6, 2022
References
P. Erdos and I. Niven, Some properties of partial sum of the harmonic series, Bull. Amer. Soc. 4 (1949), 248-251.
Q. Sun, On the number of representation of 1 by unit fraction, J. Sichuan University (in Chinese) Z1 (1978), 15-19.
Z. Cao, An Introduction to Diophantine Equations, Harbin Industrial University Press, 400, 1989.
Z. Ke, Q. Sun and X. J. Zhang, On the Diophantine equations $frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$,
J. Sichuan University (in Chinese) 3 (1964), 23-37.
Z. Ke and Q. Sun, Some conjecture and problem in number theory Chinese Journal of Nature (in Chinese) 7 (1979), 411-413.
K. Yamamoto, On the Diophantine equations $frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$, Men. Fac. Sci. Kyushu Univ. Ser. A 19 (1965), 37-47.
N. Franceschine, Egyptian fractions, Sonoma State Coll., 1978.
Palama, Sull's equarione diofantea $frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$, Boll. Union Mat. Ital. 13 (1958), 65-72.
Palama, Sull's equarione diofantea $frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$, Boll. Union Mat. Ital. 14 (1959), 82-94.
B. M. Stewart, Theory of Numbers, Macmillan, New York, 1964, pp. 198-207.
Y. Liu, On a problem of unit fraction, J. Sichuan University (in Chinese) 2 (1984), 113-114.
X. D. Yuan and J. G. Luo, On the Diophantine equation $frac{1}{x}+frac{1}{y}+frac{1}{z}=frac{1}{3 p}$, AIMS Mathematics $2(2017), 111-127$.
L. A. Rosati, Sull's equarione diofantea $frac{4}{b}=frac{1}{x_1}+frac{1}{x_2}+frac{1}{x_3}$, Boll. Union Mat. Ital. $9(1954), 59-63$.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Pushpa Publishing House, Prayagraj, India
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
