Geodesic
On the diophantine equation $x^y-y^x=c^z$
Zhongfeng Zhang 1   ; Jiagui Luo 1   ; Pingzhi Yuan 2
1 School of Mathematics and Information Science Zhaoqing University 526061 Zhaoqing, P.R. China
2 Department of Mathematics South China Normal University 510631 Guangzhou, P.R. China
Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 277-285 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Applying results on linear forms in $p$-adic logarithms, we prove that if $(x,y,z)$ is a positive integer solution to the equation $x^y-y^x=c^z$ with ${\rm gcd}(x,y)=1$ then $(x,y,z)=(2,1,k)$, $(3, 2, k)$, $k\geq 1$ if $c=1$, and either $(x,y,z)=(c^k+1,1,k)$, $k\geq 1$ or $2\leq x y\leq\max\{1.5\times 10^{10}, c\}$ if $c\geq 2$.
DOI : 10.4064/cm128-2-13
Mots-clés : applying results linear forms p adic logarithms prove positive integer solution equation y y gcd geq either geq leq leq max times geq

Zhongfeng Zhang  1   ; Jiagui Luo  1   ; Pingzhi Yuan  2

1 School of Mathematics and Information Science Zhaoqing University 526061 Zhaoqing, P.R. China
2 Department of Mathematics South China Normal University 510631 Guangzhou, P.R. China 
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     title = {On the diophantine equation $x^y-y^x=c^z$},
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Zhongfeng Zhang; Jiagui Luo; Pingzhi Yuan. On the diophantine equation $x^y-y^x=c^z$. Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 277-285. doi: 10.4064/cm128-2-13

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