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Currently displaying 1 – 2 of 2

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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)

Takafumi Miyazaki — 2014

Acta Arithmetica

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.

Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita; Takafumi Miyazaki — 2012

Colloquium Mathematicae

Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b \equiv 0 (m o d 2^{r})$ and $b \equiv \pm 2^{r} (m o d a)$ for some non-negative integer r, then the Diophantine equation $a^{x} + b^{y} = c^{z}$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).

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