On the exponential diophantine equation xy + yx = zz



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).


exponential diophantine equation upper bound for solutions singular number 

MSC 2010



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  1. [1]
    Y. Bilu, G. Hanrot, P. M. Voutier: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539 (2001), 75–122.MathSciNetMATHGoogle Scholar
  2. [2]
    G. D. Birkhoff, H. S. Vandiver: On the integral divisors of a nb n. Ann. of Math. (2) 5 (1904), 173–180.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    D. A. Buell: Computer computation of class groups of quadratic number fields. Congr. Numerantium 22 (McCarthy et al., eds.). Conf. Proc. Numerical Mathematics and Computing, Winnipeg 1978, 1979, pp. 3–12.Google Scholar
  4. [4]
    Y. Bugeaud: Linear forms in p-adic logarithms and the Diophantine equation (x n–1)/(x–1) = y q. Math. Proc. Camb. Philos. Soc. 127 (1999), 373–381.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Y. Deng, W. Zhang: On the odd prime solutions of the Diophantine equation x y + y x = z z. Abstr. Appl. Anal. 2014 (2014), Art. ID 186416, 4 pages. MR doiGoogle Scholar
  6. [6]
    M. Le: Some exponential Diophantine equations. I: The equation D 1 x 2D 2 y 2 = λk z. J. Number Theory 55 (1995), 209–221.MathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Le: On the Diophantine equation y xx y = z 2. Rocky Mt. J. Math. 37 (2007), 1181–1185.CrossRefMATHGoogle Scholar
  8. [8]
    Y. N. Liu, X. Y. Guo: A Diophantine equation and its integer solutions. Acta Math. Sin., Chin. Ser. 53 (2010), 853–856.MathSciNetMATHGoogle Scholar
  9. [9]
    F. Luca, M. Mignotte: On the equation y x ± x y = z 2. Rocky Mt. J. Math. 30 (2000), 651–661.CrossRefMATHGoogle Scholar
  10. [10]
    R. A. Mollin, H. C. Williams: Computation of the class number of a real quadratic field. Util. Math. 41 (1992), 259–308.MathSciNetMATHGoogle Scholar
  11. [11]
    L. J. Mordell: Diophantine Equations. Pure and Applied Mathematics 30, Academic Press, London, 1969.Google Scholar
  12. [12]
    A. J. van der Poorten, H. J. J. te Riele, H. C. Williams: Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100000000000. Math. Comput. 70 (2001), 1311–1328; corrig. ibid. 72 (2003), 521–523.CrossRefMATHGoogle Scholar
  13. [13]
    H. Wu: The application of BHV theorem to the Diophantine equation x y + y x = z z. Acta Math. Sin., Chin. Ser. 58 (2015), 679–684.MathSciNetMATHGoogle Scholar
  14. [14]
    Z. Zhang, J. Luo, P. Yuan: On the Diophantine equation x yy x = c z. Colloq. Math. 128 (2012), 277–285.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Z. Zhang, J. Luo, P. Yuan: On the Diophantine equation x y + y x = z z. Chin. Ann. Math., Ser. A 34A (2013), 279–284.MATHGoogle Scholar
  16. [16]
    Z. Zhang, P. Yuan: On the Diophantine equation ax y + by z + cz x = 0. Int. J. Number Theory 8 (2012), 813–821.MathSciNetCrossRefMATHGoogle Scholar

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© 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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