Yasutsugu Fujita and Maohua Le

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 3, Pages 123–129

DOI: 10.7546/nntdm.2021.27.3.123-129

**Full paper (PDF, 186 Kb)**

## Details

### Authors and affiliations

Yasutsugu Fujita

*Department of Mathematics, College of Industrial Technology, Nihon University
2-11-1 Shin-ei, Narashino, Chiba, Japan*

Maohua Le

*Institute of Mathematics, Lingnan Normal College
Zhangjiang, Guangdong, 524048 China*

### Abstract

For any positive integer , let ord denote the order of in the factorization of . Let be two distinct fixed positive integers with . In this paper, using some elementary number theory methods, the existence of positive integer solutions of the polynomial-exponential Diophantine equation with is discussed. We prove that if and ord ord, then has no solutions with . Thus it can be seen that if or , where means either and or and , then has no solutions .

### Keywords

- Polynomial-exponential Diophantine equation
- Pell’s equation

### 2020 Mathematics Subject Classification

- 11D61

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## Related papers

## Cite this paper

Fujita, Y. & Le, M. (2021). A note on the polynomial-exponential Diophantine equation (*a ^{n}* − 1)(

*b*− 1) =

^{n}*x*

^{2}.

*Notes on Number Theory and Discrete Mathematics*, 27(3), 123-129, DOI: 10.7546/nntdm.2021.27.3.123-129.