All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2)

Article Sidebar

PDF
Published: Aug 5, 2024
DOI: https://doi.org/10.60101/jarst.2023.255049
Keywords:
Diophantine equation Positive integer solution Egyptian fraction equation

Main Article Content

Suton Tadee
Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, THAILAND

Abstract

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form gif.latex?\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{u}{v} , where gif.latex?x,y,z and gif.latex?v are positive integers. Without loss of generality, we may assume that gif.latex?x\leq&space;y\leq&space;z. This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation gif.latex?\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{u}{u+1} has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation gif.latex?\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{u}{u+2}, by using elementary methods of number theory and computer calculations. In the process, we can see that gif.latex?1\leq&space;x\leq&space;9. Then, we will consider separately the value of a positive integer gif.latex?x in nine cases.  The first case is impossible. For the second and third cases, we will separate to consider the value of gif.latex?y. For the remaining cases, we will separate to consider the value of gif.latex?u. The research results showed that all positive integer solutions of the Diophantine equation gif.latex?\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{u}{u+2} are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation gif.latex?\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{u}{u+k}, where  is a positive integer with gif.latex?k\geq&space;3.

Article Details

How to Cite
1.
Tadee S. All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2). J Appl Res Sci Tech [Internet]. 2024 Aug. 5 [cited 2024 Nov. 21];23(2):255049. Available from: https://ph01.tci-thaijo.org/index.php/rmutt-journal/article/view/255049
Issue
Vol. 23 No. 2 (2024): May – August
Section
Research Articles
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

References

Rabago JFT, Tagle RP. On the area and volume of a certain regular solid and the Diophantine equation 1/2=1/x+1/y+1/z. Notes Number Theory Discrete Math. 2013;19(3):28-32.

Tadee S, Poopra S. On the Diophantine equation 1/x+1/y+1/z=1/n. Int J Math Comput Sci. 2023;18(2):173-7.

Delang L. On the equation 4/n=1/x+1/y+1/z. J Number Theory. 1981;13:485-94.

Kishan H, Rani M, Agarwal S. The Diophantine equations of second and higher degree of the form 3xy=n(x+y) and 3xyz=n(xy+yz+zx) etc. Asian J Algebra. 2011; 4(1):31-37.

Zhao W, Lu J, Wang L. On the integral solutions of the Egyptian fraction equation a/p=1/x+1/y+1/z. AIMS Math. 2021;6(5):4930-7.

Banderier C, Ruiz CAG, Luca F, Pappalardi F, Trevino E. On Egyptian fractions of length 3. Revista de La Union Mathematica Argentina. 2021;62(1):257-74.

Sandor J, Atanassov K. On a Diophantine equation arising in the history of mathematics. Notes Number Theory Discrete Math. 2021;27(1):70-5.