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)

• Autorzy: Takafumi Miyazaki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11D61: Exponential equations
• 11J86: Linear forms in logarithms; Baker's method

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 164
• Numer: 1
• Strony: 31-42

Daty

wydano

2014

Twórcy

autor

Takafumi Miyazaki

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

Identyfikatory

DOI

10.4064/aa164-1-3