Research article

On the integral solutions of the Egyptian fraction equation $ \frac ap = \frac 1x+\frac 1y+\frac 1z $

  • Received: 30 December 2020 Accepted: 19 February 2021 Published: 02 March 2021
  • MSC : 11B73, 11A07

  • It is an interesting question to investigate the integral solutions for the Egyptian fraction equation $ \frac{a}{p} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z} $, which is known as Erdős-Straus equation when $ a = 4 $. Recently, Lazar proved that this equation has not integral solutions with $ xy < \sqrt{z/2} $ and $ \gcd(x, y) = 1 $ when $ a = 4 $. But his method is difficult to get an analogous result for arbitrary $ \frac{a}{p} $, especially when $ p $ and $ a $ are lager numbers. In this paper, we extend Lazar's result to arbitrary integer $ a $ with $ 4\le a\leq\frac{1+\sqrt{1+6p^3}}{p} $, and release the condition $ \gcd(x, y) = 1 $. We show that $ \frac{a}{p} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z} $ has no integral solutions satisfying that $ xy < \sqrt{lz} $, where $ l\leq\frac{(3p+a)p}{a^2} $ when $ p\nmid y $ and $ l\leq\frac{3p^2+a}{pa^2} $ when $ p\mid y $. Besides, we extend Monks and Velingker's result to the case $ 4\le a < p $.

    Citation: Wei Zhao, Jian Lu, Lin Wang. On the integral solutions of the Egyptian fraction equation $ \frac ap = \frac 1x+\frac 1y+\frac 1z $[J]. AIMS Mathematics, 2021, 6(5): 4930-4937. doi: 10.3934/math.2021289

    Related Papers:

  • It is an interesting question to investigate the integral solutions for the Egyptian fraction equation $ \frac{a}{p} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z} $, which is known as Erdős-Straus equation when $ a = 4 $. Recently, Lazar proved that this equation has not integral solutions with $ xy < \sqrt{z/2} $ and $ \gcd(x, y) = 1 $ when $ a = 4 $. But his method is difficult to get an analogous result for arbitrary $ \frac{a}{p} $, especially when $ p $ and $ a $ are lager numbers. In this paper, we extend Lazar's result to arbitrary integer $ a $ with $ 4\le a\leq\frac{1+\sqrt{1+6p^3}}{p} $, and release the condition $ \gcd(x, y) = 1 $. We show that $ \frac{a}{p} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z} $ has no integral solutions satisfying that $ xy < \sqrt{lz} $, where $ l\leq\frac{(3p+a)p}{a^2} $ when $ p\nmid y $ and $ l\leq\frac{3p^2+a}{pa^2} $ when $ p\mid y $. Besides, we extend Monks and Velingker's result to the case $ 4\le a < p $.



    加载中


    [1] C. Elsholtz, S. Planitzer, The number of solutions of the Erdős-Straus Equation and sums of $k$ unit fractions, Proc. R. Soc. Edinburgh, 150 (2020), 1401–1427. doi: 10.1017/prm.2018.137
    [2] P. Erdős, Az $1/x_1+1/x_2+\ldots+1/x_n = a/b$ egyenlet egész számú megoldásairól, Mat. Lapok 1, (1950), 192–210.
    [3] R. Guy, Unsolved problems in number theory, 2nd ed, New York, Springer-Verlag, 1994,158–166.
    [4] R. M. Jollensten, A note on the Egyption problem, In: F. Hoffman, et al. (Eds.), Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing, Inc., Winnipeg Utilitas Math., 1976,578–589.
    [5] C. Ko, Q. Sun, S. J. Chang, On equations $4/n = 1/x+1/y+1/z$, Acta. Sci. Natur. Szechuanensis., 2 (1964), 23–37.
    [6] Y. Lazar, A remark on a conjecture of Erdős and Straus, (2020), arXiv: 2003.01237.
    [7] M. Monks, A. Velingker, On the Erdős-Straus conjecture: Properties of solutions to its underlying diophantine equation, 2008. Available from: https://pdfs.semanticscholar.org/b65e/60f1528dfc9751ee4d7b3240dd6dd8e3fbc2.pdf.
    [8] L. J. Mordell, Diophantine equations, London: Academic Press, 1969,287–290.
    [9] G. Palamà, Su di una congettura di Sierpiński relativa alla possibilità in, numeri naturali della $5/n = 1/x_1+1/x_2+1/x_3$, Boll. Un. Mat. Ital., (1958), 65–72.
    [10] B. M. Stewart, W. A. Webb, Sums of fractions with bounded numerators, Can. Math. Bull., 18 (1966), 999–1003. doi: 10.4153/CJM-1966-100-4
    [11] S. Salez, The Erdős-Straus conjecture: New modular equations and checking up to $N = 10^{17}$, (2014), arXiv: 1406.6307.
    [12] W. Sierpiński, On the decomposition of rational numbers into unit fractions (Polish), P$\acute{a}$nstwowe Wydawnictwo Nankowe, Warsaw, 1957.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1888) PDF downloads(158) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog