Solutions of Diophantine Equations Generated by Generalised Lucas Balancing Sequences

# P. Anuradha Kameswari^{1} and K. Anoosha^{1}

^{1}Department of Mathematics, Andhra University, Visakhapatnam, Andhra Pradesh, India.**Abstract:** Diophantine Equations with infinitely many solutions find their place in the applications of cryptography. In this paper we generate some quadratic Diophantine Equations in two variables with infinitely many solutions, using Generalised Lucas Balancing Sequences and obtain all the solutions of each these Diophantine Equations expressed in terms of Generalised Lucas Balancing Sequences.

**Keywords:** Diophantine equation, Lucas Balancing Sequences, Solutions.

**Cite this article as:** P. Anuradha Kameswari and K. Anoosha, *Solutions of Diophantine Equations Generated by Generalised Lucas Balancing Sequences*, Int. J. Math. And Appl., vol. 9, no. 4, 2021, pp. 75-88.

**References**

- A. Behera and G. K. Panda, On the square roots of triangular numbers, The Fib. Quart., 37(1999), 98–105.
- P. K. Ray, Balancing and cobalancing numbers, Ph.D. Thesis, Department of Mathematics, National Institute of Technology, Rourkela, India, (2009).
- P. K. Dey and S. S. Rout, Diophantine equations concerning balancing and Lucas balancing numbers, Arch. Math., 108(2017), 29–43.
- P. K. Ray, Certain matrices associated with balancing and Lucas-balancing numbers, Matematika, 28(1)(2012), 15-22.
- G. K. Panda and S. S. Rout, A class of recurrent sequences exhibiting some exciting properties of balancing numbers, Int. J. Math. Comp. Sci. Eng., 6(2012), 4–6.
- G. K. Panda and P. K. Ray, Some links of balancing and cobalancing numbers with Pell andassociated Pell numbers, Bull. Inst. Math. Acad. Sin. (N. S.), 6(2011), 41-72.
- A. Marlewski and Piotr Zarzycki, Infinitely many positive solutions of the Diophantine equation $x ^2 -kxy+y^2 +x=0$, Computers and Mathematics with Applications, 47(2004), 10.1016/S0898-1221(04)90010-7.
- Mahyar Bahramian and Hassan Daghigh, A generalized Fibonacci sequence and the Diophantine equations $x^2 \pm kxy - y^2 \pm x = 0$, Iranian Journal of Mathematical Sciences and Informatics, 8(2013), 111-121.
- Julian Rosen, Krishnan Shankar and Justin Thomas, Square Roots, Continued Fractions and the Orbit of $1/0$ on $H^2$ (2006).
- I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley and Sons, New York, N, USA, 5th edition, (1991).