On the Transcendental Equation With Three Unknowns $\sqrt{2z-4}=\sqrt{x+\sqrt{C}y}\pm \sqrt{x-\sqrt{C}y}$ for Different Values of C by Using the Continued Fraction Method

# S. Sriram^{1} and P. Veeramallan^{1,2}

^{1}Department of Mathematics, National College (Affiliated to Bharathidasan University), Tiruchirappalli, Tamilnadu, India.

^{2}P.G Assistant in Mathematics, GHSS, Yethapur, Salem, Tamilnadu, India.**Abstract:** The Transcendental equation with three unknowns is given by
$\sqrt{2z-4}=\sqrt{x+\sqrt{C}y}\pm \sqrt{x-\sqrt{C}y}$ is considered and analyzed for finding a different set of integer solutions utilizingthe continued fraction method, under numerous patterns with some numerical examples.

**Keywords:** Transcendental equation, Integer solutions, Continued fraction, Pell equation, Recurrence relation.

**Cite this article as:** S. Sriram and P. Veeramallan, *On the Transcendental Equation With Three Unknowns $\sqrt{2z-4}=\sqrt{x+\sqrt{C}y}\pm \sqrt{x-\sqrt{C}y}$ for Different Values of C by Using the Continued Fraction Method*, Int. J. Math. And Appl., vol. 10, no. 1, 2022, pp. 51-57.

**References**

- L. E. Dickson, History of Theory of Numbers, Vol. II, Chelsea PublishingCompany, New York, (1952).
- D. E. Smith, History of Mathematics, Vol. I and II, Dover Publications, New York, (1953).
- David M. Burton, Elementary Number Theory, Seventh Edition, TheMcGraw Hill Companies, New York, (2011).
- Titu Andreescu, Dorin Andrica and Ion Cucurezeanu, n introduction to Diophantine Equations, Birhauser, New York, (2010).
- Ahmet Tekcan, Continued Fractions Expansions of $\sqrt{D}$ and Pell Equation $x^2-Dy^2=1$, Mathematica Moravica, 15(2)(2011), 19-27.
- M. A. Gopalan, P. Shanmuganandham and S. Sriram, On Transcendental Equation $z=\sqrt{x+\sqrt{B}y}+\sqrt{x-\sqrt{B}y}$, Antartica Journal of Mathematics, 7(5)(2010), 509-515.