Qiongzhi Tang

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 2, Pages 88–100

DOI: 10.7546/nntdm.2021.27.2.88-100

**Full paper (PDF, 204 Kb)**

## Details

### Authors and affiliations

Qiongzhi Tang

*School of Mathematics and Statistics, Changsha University of Science and Technology,
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering
Changsha, 410114, China
*

### Abstract

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations for certain polynomials , which mean to construct integral triangles with two sides given by the values of polynomials and with the intersection angle or .

### Keywords

- Diophantine equation
- Pell equation
- Positive integer solution

### 2020 Mathematics Subject Classification

- 11D25
- 11D72

### References

- Burn, B. (2003). Triangles with a 60◦ angle and sides of integer length. The Mathematical Gazette, 87(508), 148–153.
- Dickson, L. E. (2005). History of the Theory of Numbers, Vol. II: Diophantine Analysis. Dover Publications.
- Gilder, J. (1982). Integer-sided triangles with an angle of 60◦. The Mathematical Gazette, 66(438), 261–266.
- He, B., Togbé, A., & Ulas, M. (2010). On the Diophantine equation z
^{2}= f(x)^{2}± f(y)^{2}, II. Bulletin of the Australian Mathematical Society, 82(2), 187–204. - Pocklington, H. C. (1914). Some Diophantine impossibilities. Mathematical Proceedings of the Cambridge Philosophical Society, 17, 110–118.
- Read, E. (2006). On integer-sided triangles containing angles of 120◦ or 60◦. The Mathematical Gazette, 90(518), 299–305.
- Selkirk, K. (1983). Integer-sided triangles with an angle of 120◦. The Mathematical Gazette, 67(442), 251–255.
- Sierpinski, W. (1962). Triangular Numbers. Biblioteczka Matematyczna 12, Warszawa (in Polish).
- Tengely, Sz., & Ulas, M. (2017). On certain Diophantine equations of the form z
^{2}= f(x)^{2}±g(y)^{2}. Journal of Number Theory, 174, 239–257. - Ulas, M., & Togbé, A. (2010). On the Diophantine equation z
^{2}= f(x)^{2}± f(y)^{2}.

Publicationes Mathematicae Debrecen, 76(1–2), 183–201. - Youmbai, A. E. A., & Behloul, D. (2019). Rational solutions of the Diophantine equations f(x)
^{2}± f(y)^{2}= z^{2}. Periodica Mathematica Hungarica, 79(2), 255–260. - Zhang, Y., & Tang, Q. Z. (2021). On the integer solutions of the Diophantine equations z2 = f(x)
^{2}± f(y)^{2}. Periodica Mathematica Hungarica, accepted. - Zhang, Y., Tang, Q. Z., & Zhang, Y. N. (2020). On the Diophantine equations z
^{2}= f(x)^{2}± f(y)^{2}involving Laurent polynomials, II. submitted. - Zhang, Y., & Zargar, A. S. (2019). On the Diophantine equations z
^{2}= f(x)^{2}± f(y)^{2}

involving quartic polynomials. Periodica Mathematica Hungarica, 79(1), 25–31. - Zhang, Y., & Zargar, A. S. (2020). On the Diophantine equations z
^{2}= f(x)^{2}± f(y)^{2}involving Laurent polynomials. Functiones et Approximatio, Commentarii Mathematici, 62(2), 187–201.

## Related papers

## Cite this paper

Tang, Q. (2021). On the Diophantine equations *z*^{2} = *f*(*x*)^{2} ± *f*(*x*)*f*(*y*) + *f*(*y*)^{2}. *Notes on Number Theory and Discrete Mathematics*, 27(2), 88-100, DOI: 10.7546/nntdm.2021.27.2.88-100.