Geodesic
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)
Takafumi Miyazaki1
1 Department of Mathematics College of Science and Technology Mathematics Department and Information Sciences Nihon University Suruga-Dai Kanda, Chiyoda Tokyo 101-8308, Japan
Acta Arithmetica, Tome 164 (2014) no. 1, pp. 31-42 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa164-1-3
Keywords: positive integers even integers satisfying sqrt sqrt prove diophantine equation has positive integer solutions other whenever result explicit refinement theorem due luca

Takafumi Miyazaki 1

1 Department 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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(Acta Arith. 153 (2012), 373–392)
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“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)
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Takafumi Miyazaki. 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). Acta Arithmetica, Tome 164 (2014) no. 1, pp. 31-42. doi: 10.4064/aa164-1-3

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