**Ahmet Tekcan**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 1, Pages 108–121

DOI: 10.7546/nntdm.2019.25.1.108-121

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

## Details

### Authors and affiliations

Ahmet Tekcan

*Bursa Uludag University, Faculty of Science
Department of Mathematics, Bursa, Turkey
*

### Abstract

In this work, we derive some new algebraic relations on all almost balancing numbers (of first and second type) and triangular (and also square triangular) numbers.

### Keywords

- Balancing numbers
- Almost balancing numbers
- Triangular numbers
- Square triangular numbers

### 2010 Mathematics Subject Classification

- 11B37
- 11B39

### References

- Bugeaud, Y. (1999). Linear forms in
*p*-adic logarithms and the Diophantine equation (*x*^{n}– 1)/(*x*– 1) =*y*.^{q}*Math. Proc. Cambridge Phil. Soc.*, 127 (3), 373–381. - Cao, Z. & Dong, X. (2002). On the Terai-Jeśmanowicz conjecture.
*Publ. Math. Debrecen*, 61 (3–4), 253–265. - Cao, Z. & Dong, X. (2003). An application of a lower bound for linear forms in two logarithms to the Terai-Jeśmanowicz conjecture.
*Acta Arith.*, 110 (2), 153–164. - Cipu, M. & Mignotte, M. (2009). On a conjecture on exponential Diophantine equations.
*Acta Arith.*, 140 (3), 251–269. - Jeśmanowicz, L. (1955/56). Several remarks on Pythagorean numbers.
*Wiadom. Mat.,*1 (2), 196–202 (in Polish). - Laurent, M. (2008). Linear forms in two logarithms and interpolation determinants II.
*Acta Arith*., 133 (4), 325–348. - Laurent, M., Mignotte, M. & Nesterenko, Y. (1995). Formes linéaires en deux logarithmes et déterminants dínterpolation.
*J. Number Theory*, 55 (2), 285–321. - Le, M. (2003). A conjecture concerning the exponential Diophantine equation
*a*+^{x}*b*=^{y}*c*.^{z}*Acta Arith.*, 106 (4), 345–353. - Le, M., Togbe, A. & Zhu, H. (2014). On a pure ternary exponential Diophantine equation.
*Publ. Math. Debrecen*, 85 (3–4), 395–411. - Lu, W. (1959). On the Pythagorean numbers 4
*n*^{2}– 1, 4*n*and 4*n*^{2}+ 1.*Acta Sci. Natur. Univ. Szechuan*, 2, 39–42 (in Chinese). - Luca, F. (2012). On the system of Diophantine equations
*a*^{2}+*b*^{2}= (*m*^{2}+ 1)^{r}and*a*+^{x}*b*= (^{y}*m*^{2}+ 1).^{z}*Acta Arith.*, 153 (4), 373–392. - Miyazaki, T. (2011). Terai’s conjecture on exponential Diophantine equations.
*Int. J. Number Theory*, 7 (4), 981–999. - Miyazaki, T. (2014). A note on the article by F. Luca “On the system of Diophantine equations
*a*^{2}+*b*^{2}= (*m*^{2}+ 1)^{r}and*a*+^{x}*b*= (^{y}*m*^{2}+ 1)” (Acta Arith. 153 (2012), 373–392).^{z}*Acta Arith*., 164 (1), 31–42. - Scott, R. & Styer, R. (2016). Number of solutions to
*a*+^{x}*b*=^{y}*c*.^{z}*Publ. Math. Debrecen*, 88 (1–2), 131–138. - Terai, N. (1994). The Diophantine equation
*a*+^{x}*b*=^{y}*c*.^{z}*Proc. Japan Acad. Ser. A Math. Sci.*, 70 (1), 22–26. - Terai, N. (1995). The Diophantine equation
*a*+^{x}*b*=^{y}*c*II.^{z}*Proc. Japan Acad. Ser. A Math. Sci*., 71 (6), 109–110. - Terai, N. (1996). The Diophantine equation
*a*+^{x}*b*=^{y}*c*III.^{z}*Proc. Japan Acad. Ser. A Math. Sci*., 72 (1), 20–22. - Terai, N. (1999). Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations.
*Acta Arith.*, 90 (1), 17–35.

## Related papers

- Rayaguru, S. G. & Panda, G. K. (2020). A generalization to almost balancing and cobalancing numbers using triangular numbers.
*Notes on Number Theory and Discrete Mathematics*, 26(3), 135-148. - Tekcan, A., & Türkmen, E. Z. (2023). Almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers.
*Notes on Number Theory and Discrete Mathematics*, 29(4), 682-694.

## Cite this paper

Tekcan, A. (2019). Almost balancing, triangular and square triangular numbers. *Notes on Number Theory and Discrete Mathematics*, 25(1), 108-121, DOI: 10.7546/nntdm.2019.25.1.108-121.