A note on the article by F. Luca “On the system of Diophantine equations $a²+b² = (m²+1)^r$ and $a^{x}+b^y = (m²+1)^z$” (Acta Arith. 153 (2012), 373-392)

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)

Takafumi Miyazaki

Acta Arithmetica, Tome 166 (2014), p. 31-42 / Harvested from The Polish Digital Mathematics Library

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Résumé

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.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:279513

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     author = {Takafumi Miyazaki},
     title = {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)},
     journal = {Acta Arithmetica},
     volume = {166},
     year = {2014},
     pages = {31-42},
     language = {en},
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Takafumi Miyazaki. A note on the article by F. Luca “On the system of Diophantine equations $a²+b² = (m²+1)^r$ and $a^{x}+b^y = (m²+1)^z$” (Acta Arith. 153 (2012), 373-392). Acta Arithmetica, Tome 166 (2014) pp. 31-42. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-1-3/