On Terai’s exponential equation with two finite integer parameters

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
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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 = (m2 + 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

References

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Cite this paper

APA

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.

Chicago

Miyazaki, Takafumi. “On Terai’s Exponential Equation with Two Finite Integer Parameters.” Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 84-107, doi: 10.7546/nntdm.2019.25.1.84-107.

MLA

Miyazaki, Takafumi. “On Terai’s Exponential Equation with Two Finite Integer Parameters.” Notes on Number Theory and Discrete Mathematics 25.1 (2019): 84-107. Print, doi: 10.7546/nntdm.2019.25.1.84-107.

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