Takafumi Miyazaki

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 1, Pages 84–107

DOI: 10.7546/nntdm.2019.25.1.84-107

**Full paper (PDF, 247 Kb)**

## Details

### Authors and affiliations

Takafumi Miyazaki

*Division of Pure and Applied Science
Faculty of Science and Technology
Gunma University
1-5-1 Tenjin-cho, Kiryu, Gunma, Japan
*

### Abstract

Let *r* be an integer with *r* > 1, and *m* be an even positive integer. Define integers *A* and *B* by the equation *A* + *B* √−1 = (*m* + √−1)^{r}. It is proven by F. Luca in 2012 that the equation |*A*|^{x} + |*B*|^{y} = (*m*^{2} + 1)^{z} does not hold for any triple (*x*, *y*, *z*) of positive integers not equal to (2, 2, *r*), whenever *r* or *m* exceeds some effectively computable absolute constant. In our previous work, we estimated this constant explicitly. Here that estimate is substantially improved.

### Keywords

- Exponential Diophantine equation
- Linear forms in logarithms of algebraic numbers

### 2010 Mathematics Subject Classification

- 11D61
- 11J86

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## Cite this paper

Miyazaki, T. (2019). On Terai’s exponential equation with two finite integer parameters. *Notes on Number Theory and Discrete Mathematics*, 25(1), 84-107, DOI: 10.7546/nntdm.2019.25.1.84-107.