On the exponential diophantine equation xy + yx = zz

• Published in: Czechoslovak mathematical journal Vol. 67; no. 3; pp. 645 - 653
• Main Author: Du, Xiaoying
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2017
• Subjects: Analysis; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; upper bound for solutions; singular number; 11D61; exponential diophantine equation
• ISSN: 0011-4642; 1572-9141
• DOI: 10.21136/CMJ.2017.0645-15

For any positive integer D which is not a square, let ( u 1 , v 1 ) be the least positive integer solution of the Pell equation u 2 − Dv 2 = 1, and let h (4 D ) denote the class number of binary quadratic primitive forms of discriminant 4 D . If D satisfies 2 ł D and v 1 h(4 D ) ≡ 0 (mod D ), then D is called a singular number. In this paper, we prove that if ( x, y, z ) is a positive integer solution of the equation x y + y x = z z with 2 | z , then maximum max{ x, y, z } <480000 and both x, y are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions ( x, y, z ).