A note on the exponential diophantine equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$
Colloquium Mathematicum, Tome 149 (2017) no. 2, pp. 265-273
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
Let $a, b, c, m$ be positive integers such that $a+b=c^{2}$, $2\nmid c$, $m \gt 1$ and $m\equiv \pm 1\ ({\rm mod}\,c)$. We prove that if $a\equiv 4$ or $5\ ({\rm mod}\, 8)$, $((a+1)/c)=-1$ and $m \gt 6c^2\log c$, where $((a+1)/c)$ is the Jacobi symbol, then the equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$ only has the positive integer solution $(x, y, z)=(1, 1, 2)$.
Keywords:
positive integers nmid equiv mod prove equiv mod log where jacobi symbol equation only has positive integer solution
Affiliations des auteurs :
Xiaowei Pan 1
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author = {Xiaowei Pan},
title = {A note on the exponential diophantine equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$},
journal = {Colloquium Mathematicum},
pages = {265--273},
publisher = {mathdoc},
volume = {149},
number = {2},
year = {2017},
doi = {10.4064/cm6878-10-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm6878-10-2016/}
}
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Xiaowei Pan. A note on the exponential diophantine equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$. Colloquium Mathematicum, Tome 149 (2017) no. 2, pp. 265-273. doi: 10.4064/cm6878-10-2016
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