Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

Abstract

The Ternary cubic Diophantine Equation represented by6(x^2+y^2 )-11xy=288z^3 is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed.

Introduction

Mathematics is the language of patterns and relationships and is used to describe anything that can be quantified.Diophantine equations have stimulated the interest of various mathematicians. Diophantine equations with higher degree greater than three can be reduced in to equations of degree 2 or 3 and it can be easily solved.In [1-3], theory of numbers is discussed.  In [4-5], quadratic Diophantine equations are discussed. In [6-11], cubic, biquadratic and higher order equations are considered for its integral solutions. In this communication the non-homogeneous cubic equation with three unknowns represented by the equation  is considered and in particular a few interesting relations among the solutions are presented.

A. Notations

II. METHOD OF ANALYSIS

The ternary cubic Diophantine equation to be solved for its non-zero integral solutions is

The substitution of linear transformations

In (1) leads to,

A. Pattern: 1

Assume,

where a  and b  are non-zero integers.

Using (4) and (5) in (3), and employing the method of factorization

Equating the like terms and comparing the rational and irrational parts, we get

Substituting the above values of  u & v  in equation (2), the corresponding integer solutions of (1) are given by

Observations

B. Pattern: 2

Instead of (5), we write 288 as

Using (4) and (7) in (3), and employing the method of factorization,

Equating the like terms and comparing the rational and irrational parts, we get

 Substituting the above values of  u & v  in equation (2), the corresponding integer solutions of (1) are given by

Observations

C. Pattern: 3

288 in (3) can be written as

Using (4) & (9) in (3),

Equating the like terms and comparing the real and imaginary parts, we get

As our intension is to find integer solutions, we suitably choosea=3A and b=3B ,then the values of

Observations

Conclusion

In this paper we have presented three different patterns of non-zero distinct integers solutions of the non-homogeneous cone given by6(x^2+y^2 )-11xy=288z^3.To conclude one may search for other patterns of non-zero integer distinct solutions and their corresponding properties for other choices of cubic Diophantine equations.

References

[1] Carmichael, R.D., The theory of numbers and Diophantine Analysis, Dover Publications,New York, 1959. [2] Dickson L.E, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York,1952. [3] Mordell. L.J, Diophantine equations, Academic Press,London,1969 Telang, S.G., Number theory, Tata McGraw Hill publishing company, New Delhi, 1996. [4] Gopalan.M.A.,Vidhyalakshmi.S and Umarani.J., “On ternary Quadratic Diophantine equation ”, Sch.J. Eng. Tech. 2(2A); 108- 112,2014. [5] Janaki.G and Saranya.C.,Observations on the Ternary Quadratic Diophantine Equation , International Journal of Innovative Research in Science, Engineering and Technology, Vol-5, Issue-2, Pg.no: 2060-2065, Feb 2016. [6] Janaki.G and Vidhya.S., On the integer solutions of thehomogeneous biquadratic Diophantine equation , International Journal of Engineering Science and Computing, Vol. 6, Issue 6, pp.7275-7278, June, 2016. [7] Gopalan.M.A and Janaki.G, Integral solutions of , Impact J.Sci.,Tech., 4(1), 97-102, 2010. [8] Janaki.G and Saranya.P., On the ternary Cubic Diophantineequation , International Journal of Science and Research- online, Vol 5, Issue 3, Pg.No:227-229, March 2016. [9] Janaki.G and Saranya.C., Integral Solutions of the non-homogeneous heptic equation with five unknowns , International Journal of Engineering Science and Computing, Vol. 6, Issue 5, pp.5347-5349, May, 2016. [10] Janaki.G and Saranya.C., Integral Solutions of the ternary cubic equation , International Research Journal of Engineering and Technology, Vol. 4, Issue 3, pp.665-669, March, 2017. [11] Janaki.G and Saranya.C., Integral Solutions of the homogeneous biquadratic Diophantine equation , International Journal for Research in Applied Science and Engineering Technology, Vol. 5, Issue 8, pp.1123-1127, Aug 2017.

Copyright

Copyright © 2022 C. Saranya, P. Kayathri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.