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Keywords:
exponential Diophantine equation; sieving; modular computations
Summary:
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.
References:
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