Geodesic
A note on ternary purely exponential diophantine equations
Yongzhong Hu 1   ; Maohua Le 2
1 Department of Mathematics Foshan University 528000 Foshan, Guangdong, P.R. China
2 Institute of Mathematics Lingnan Normal University 524048 Zhanjiang, Guangdong, P.R. China
Acta Arithmetica, Tome 171 (2015) no. 2, pp. 173-182 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$, and let $m=\max \{a,b,c\}$. Using the Gel'fond–Baker method, we prove that all positive integer solutions $(x,y,z)$ of the equation $a^x+b^y=c^z$ satisfy $\max \{x,y,z\}155000(\log m)^3$. Moreover, using that result, we prove that if $a,b,c$ satisfy certain divisibility conditions and $m$ is large enough, then the equation has at most one solution $(x,y,z)$ with $\min\{x,y,z\}>1$.
DOI : 10.4064/aa171-2-4
Keywords: fixed coprime positive integers min max using gelfond baker method prove positive integer solutions equation y satisfy max log moreover using result prove satisfy certain divisibility conditions large enough equation has solution min

Yongzhong Hu  1   ; Maohua Le  2

1 Department of Mathematics Foshan University 528000 Foshan, Guangdong, P.R. China
2 Institute of Mathematics Lingnan Normal University 524048 Zhanjiang, Guangdong, P.R. China 
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Yongzhong Hu; Maohua Le. A note on ternary purely exponential diophantine equations. Acta Arithmetica, Tome 171 (2015) no. 2, pp. 173-182. doi: 10.4064/aa171-2-4

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