Complete solution of the Diophantine equation $x^y+y^x=z^z

Complete solution of the Diophantine equation $x^y+y^x=z^z

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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Veröffentlicht in:Czechoslovak Mathematical Journal 2019-07, Vol.69 (2), p.479-484
1. Verfasser: Cipu, Mihai
Format: Artikel
Sprache:eng
Schlagworte:
Diophantine equation
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Zusammenfassung: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.
ISSN:0011-4642
1572-9141
DOI:10.21136/CMJ.2018.0395-17