Research article

On the Diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3p}$

  • Received: 04 December 2016 Accepted: 06 January 2017 Published: 24 February 2017
  • In the present paper we obtained all positive integer solutions of some diophantine equations related to unit fraction.

    Citation: Xiaodan Yuan, Jiagui Luo. On the Diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3p}$[J]. AIMS Mathematics, 2017, 2(1): 111-127. doi: 10.3934/Math.2017.1.111

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  • In the present paper we obtained all positive integer solutions of some diophantine equations related to unit fraction.


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    [1] Z. Cao, An intruduction to Diophantine equations, Haerbin Industril university Press, 400 (1989).
    [2] N. Franceschine, Egyptian fractions, Sonoma State Coll., 1978.
    [3] Z. Ke, Q. Sun, X. Zhang, On the Diophantine equatins $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, J. Sichuan University (in Chinese), 3 (1964), 23-37.
    [4] Z. Ke, Q. Sun, Some conjecture and problem in Number Theorey, Chinese Journal of Nature (in Chinese), 7 (1979), 411-413.
    [5] Y. Liu, On a problem of unit fraction, J. Sichuan University (in Chinese), 2 (1984), 113-114.
    [6] 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.
    [7] 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.
    [8] 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.
    [9] B.M.Stewart, Theory of numbers, Macmillan, New York, (1964), 198-207.
    [10] Yamamoto, On the Diophantine equatins $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, K. Men. Fac. Sci. Kyushu University, Ser.A., 19 (1965), 37-47.
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  • © 2017 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)
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