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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)

100%

Miyazaki T.

Acta Arithmetica

2014

tom 164

nr 1

31-42

Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A + B√(-1) = (m + √(-1))^{r}$. We prove that the Diophantine equation $|A|^x + |B|^y = (m² + 1)^z$ 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

63%

Fujita Y., Miyazaki T.

Colloquium Mathematicum

2012

tom 128

nr 2

211-222

Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b ≡ 0 (mod 2^{r})$ and $b ≡ ±2^{r} (mod a)$ 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).

Ograniczanie wyników

1 Acta Arithmetica

1 Colloquium Mathematicum

2 Miyazaki T.

1 Fujita Y.

1 2014

1 2012

Wyniki wyszukiwania

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)

Jeśmanowicz' conjecture with congruence relations