On the exponential diophantine equation xy + yx = zz

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Abstract

For any positive integer D which is not a square, let (u1, v1) be the least positive integer solution of the Pell equation u2Dv2 = 1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2 ł D and v1h(4D) ≡ 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 xy + yx = zz 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).

Keywords

exponential diophantine equation upper bound for solutions singular number 

MSC 2010

11D61 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.School of MathematicsJinzhong UniversityJinzhong, ShanxiP.R. China

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