Abstract
The triples (x, y, z) = (1,zz − 1,z), (x, y, z) = (zz − 1,1,z), where z ∈ ℕ, satisfy the equation xy + yx = zz. In this paper it is shown that the same equation has no integer solution with min{x,y,z} > 1, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.
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References
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.
Y. Deng, W. Zhang: On the odd prime solutions of the Diophantine equation x y + y x = z z. Abstr. Appl. Anal. 2014 (2014), Article ID 186416, 4 pages.
X. Du: On the exponential Diophantine equation x y + yx = z z. Czech. Math. J. 67 (2017), 645–653.
R. A. Mollin, H. C. Williams: Computation of the class number of a real quadratic field. Util. Math. 41 (1992), 259–308.
The PARI-Group: PARI/GP version 2.3.5. Univ. Bordeaux, 2010; Available at https://doi.org/pari.math.u-bordeaux.fr/download.html.
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; corrigenda and addition ibid. 72 (2003), 521–523.
H. M. Wu: The application of the BHV theorem to the Diophantine equation x y + y x = z z. Acta Math. Sin., Chin. Ser. 58 (2015), 679–684. (In Chinese.)
Z. Zhang, J. Luo, P. Yuan: On the Diophantine equation x y − y x = z z. Chin. Ann. Math., Ser. A 34 (2013), 279–284. (In Chinese.)
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Cipu, M. Complete solution of the diophantine equation xy + yx = zz. Czech Math J 69, 479–484 (2019). https://doi.org/10.21136/CMJ.2018.0395-17
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DOI: https://doi.org/10.21136/CMJ.2018.0395-17