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 |
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| container_title | Czechoslovak Mathematical Journal |
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| creator | Cipu, Mihai |
| description | 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. |
| doi_str_mv | 10.21136/CMJ.2018.0395-17 |
| format | Article |
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| ispartof | Czechoslovak Mathematical Journal, 2019-07, Vol.69 (2), p.479-484 |
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| source | EZB-FREE-00999 freely available EZB journals; Alma/SFX Local Collection; SpringerLink Journals - AutoHoldings |
| subjects | Diophantine equation |
| title | Complete solution of the Diophantine equation $x^y+y^x=z^z |
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