Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 479-484, 2019
Complete solution of the Diophantine equation $x^y+y^x=z^z$
Mihai Cipu
Received August 25, 2017. Published online August 7, 2018.
Abstract: The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in\Bbb N$, satisfy the equation $x^y+y^x=z^z$. 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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Affiliations: Mihai Cipu, Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit nr. 5, P.O. Box 1-764, RO-014700 Bucharest, Romania, e-mail: Mihai.Cipu@imar.ro
