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

**Download 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

### References

- Bennett, M. A., & Skinner, C. M. (2004). Ternary Diophantine equations via Galois representations and modular forms. Canadian Journal of Mathematics, 56, 23–54.
- Cohn, J. H. E. (2002). The Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Periodica Mathematica Hungarica, 44, 169–175. - Guo, X.-Y. (2013). A note on the Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Periodica Mathematica Hungarica, 66, 87–93. - Hajdu, L., & Szalay, L. (2000). On the Diophantine equations (2
− 1)(6^{n}− 1) = x^{n}^{2}and (an − 1)(akn − 1) = x2. Periodica Mathematica Hungarica, 40, 141–145. - Ishii, K. (2016). On the exponential Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Publicationes Mathematicae Debrecen, 89, 253–256. - Le, M.-H. (2009). A note on the exponential Diophantine equation (2
− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Publicationes Mathematicae Debrecen, 74, 401–403. - Le, M.-H., & Soydan, G. (2020). A brief survey on the generalized Lebesgue–Ramanujan–Nagell equation. Surveys in Mathematics and its Applications, 15, 473–523.
- Li, L., & Szalay, L. (2010). On the exponential Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Publicationes Mathematicae Debrecen, 77, 465–470. - Li, Z.-J. (2011). Research for the solution of the Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Master’s thesis, Wuhu: Anhui Normal University (in Chinese). - Li, Z.-J., & Tang, M. (2010). On the Diophantine equation (2
− 1)(^{n}*a*− 1) = x2. Journal of Anhui Normal University, Natural Science, 33, 515–517 (in Chinese).^{n} - Li, Z.-J., & Tang, M. (2011). A remark on a paper of Luca and Walsh. Integers, 11, 827–832.
- Liang, M. (2012). On the Diophantine equation (
*a*− 1)((^{n}*a*+ 1)− 1) =^{n}*x*^{2}. Journal of Mathematics (Wuhan), 32, 511–514 (in Chinese). - Luca, F., & Walsh, P. G. (2002). The product of like-indexed terms in binary recurrences. Journal of Number Theory, 96, 152–173.
- Mordell, L. J. (1969). Diophantine Equations, London: Academic Press.
- Noubissie, A., Togbe, A., & Zhang, Z.-F. (2020). On the exponential Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. The Bulletin of the Belgian Mathematical Society, 27, 161–166. - Szalay, L. (2000). On the Diophantine equation (2
− 1)(3^{n}− 1) =^{n}*x*^{2}. Publicationes Mathematicae Debrecen, 57, 1–9. - Tang, M. (2011). A note on the exponential Diophantine equation (
*a*− 1)(^{m}*b*− 1) =^{n}*x*^{2}. Journal of Mathematical Research and Exposition, 31, 1064–1066. - Yang, H., Pei, Y.-T., & Fu, R.-Q. (2016). The solvability of the Diophantine equation (
*a*− 1)((^{n}*a*+ 1)− 1) =^{n}*x*^{2}. Journal of Xiamen University, Natural Science, 55, 91–93 (in Chinese). - Yang, S.-C., Wu, W.-Q., & Zheng, H. (2011). The solution of the Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Journal of Southwest University, Natural Science, 37, 31–34 (in Chinese). - Yuan, P.-Z., & Zhang, Z.-F. (2012). On the Diophantine equation (
*a*− 1)(^{n}*b*− 1) =^{n}*x*^{2}. Publicationes Mathematicae Debrecen, 80, 327–331.

## 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.