Geodesic
Complete solution of the Diophantine equation $x^y+y^x=z^z$
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 479-484 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/CMJ.2018.0395-17
Classification : 11A15, 11D61
Keywords: exponential Diophantine equation; sieving; modular computations 
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     author = {Cipu, Mihai},
     title = {Complete solution of the {Diophantine} equation $x^y+y^x=z^z$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {479--484},
     year = {2019},
     volume = {69},
     number = {2},
     doi = {10.21136/CMJ.2018.0395-17},
     mrnumber = {3959960},
     zbl = {07088800},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0395-17/}
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Cipu, Mihai. Complete solution of the Diophantine equation $x^y+y^x=z^z$. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 479-484. doi: 10.21136/CMJ.2018.0395-17

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