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A note on the article by F. Luca “On the system of Diophantine equations $a^2+b^2=(m^2+1)^r$ and $a^{x}+b^y=(m^2+1)^z$” (Acta Arith. 153 (2012), 373–392)

Volume 164 / 2014

Takafumi Miyazaki Acta Arithmetica 164 (2014), 31-42 MSC: Primary 11D61; Secondary 11J86. DOI: 10.4064/aa164-1-3


Let $r,m$ be positive integers with $r>1$, $m$ even, and $A,B$ be integers satisfying $A+B \sqrt {-1}=(m+\sqrt {-1})^{r}$. We prove that the Diophantine equation $|A|^x+|B|^y=(m^{2}+1)^z$ has no positive integer solutions in $(x,y,z)$ other than $(x,y,z)=(2,2,r)$, whenever $r>10^{74}$ or $m>10^{34}$. Our result is an explicit refinement of a theorem due to F. Luca.


  • Takafumi MiyazakiDepartment of Mathematics
    College of Science and Technology
    Mathematics Department and Information Sciences
    Nihon University
    Suruga-Dai Kanda, Chiyoda
    Tokyo 101-8308, Japan

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