Czechoslovak Mathematical Journal, first online, pp. 1-6


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.
Keywords:  exponential Diophantine equation; sieving; modular computations
Classification MSC:  11D61, 11A15
DOI:  10.21136/CMJ.2018.0395-17

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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


 
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