**R. K. Davala**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 29, 2023, Number 3, Pages 495–502

DOI: 10.7546/nntdm.2023.29.3.495-502

**Full paper (PDF, 222 Kb)**

## Details

### Authors and affiliations

R. K. Davala

*VIT-AP University, Amaravati, Andhra Pradesh, India
*

### Abstract

Let and be the -th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations and for and present the non-negative integer solutions of the Diophantine equations in each case.

### Keywords

- Balancing numbers
- Lucas-balancing numbers
- Diophantine equation

### 2020 Mathematics Subject Classification

- 11B39
- 11B37

### References

- Behera, A., & Panda, G. K. (1999). On the square roots of triangular numbers.
*The Fibonacci Quarterly*, 37(2), 98–105. - Beiter, M. (1964). The midterm coefficient of the cyclotomic polynomial .
*American Mathematical Monthly*, 71, 769–770. - Chu, H. V. (2020). Representation of and .
*The Fibonacci Quarterly*, 58(4), 334–339. - Davala, R. K. (2015). On convolution and binomial sums of balancing and Lucas-balancing numbers.
*International Journal of Mathematical Sciences and Engineering Applications*, 8(5), 77–83. - Davala, R. K. (2018).
*Algebraic and geometric aspects of some binary recurrence*

*sequences*. Ph.D. Thesis, National Institute of Technology, Rourkela, India. - Davala, R. K., & Panda, G. K. (2015). On sum and ratio formulas for balancing numbers.
*The Journal of the Indian Mathematical Society*, 82(2), 23–32. - Davala, R. K., & Panda, G. K. (2016). On sum and ratio formulas for balancing-like sequences.
*Notes on Number Theory and Discrete Mathematics*, 22(3), 45–53. - Davala, R. K., & Panda, G. K. (2019). On sum and ratio formulas for Lucas-balancing numbers.
*Palestine Journal of Mathematics*, 8(2), 200–206. - Frontczak, R. (2018). Sums of balancing and Lucas-balancing numbers with binomial coefficients.
*International Journal of Mathematical Analysis*, 12, 585–594. - Panda, G. K. (2009). Some fascinating properties of balancing numbers.
*Congressus Numerantium*, 194, 265–271. - Ray, P. K. (2009).
*Balancing and cobalancing numbers*. Ph.D. Thesis, National Institute of Technology, Rourkela, India. - Ray, P. K. (2015). Balancing and Lucas-balancing sums by matrix methods.
*Mathematical Reports*, 17(2), 225–233. - Soykan, Y. (2021). A study on generalized balancing numbers.
*Asian Journal of Advanced Research and Reports*, 15(5), 78–100.

### Manuscript history

- Received: 10 March 2023
- Revised: 28 June 2023
- Accepted: 13 July 2023
- Online First: 15 July 2023

### Copyright information

Ⓒ 2023 by the Author.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

- Davala, R. K., & Panda, G. K. (2016). On sum and ratio formulas for balancing-like sequences.
*Notes on Number Theory and Discrete Mathematics*, 22(3), 45–53.

## Cite this paper

Davala, R. K. (2023). Solution to a pair of linear, two-variable, Diophantine equations with coprime coefficients from balancing and Lucas-balancing numbers. *Notes on Number Theory and Discrete Mathematics*, 29(3), 495-502, DOI: 10.7546/nntdm.2023.29.3.495-502.