On the Diophantine Equation

London Journal of Research in Science: Natural and Formal
Volume | Issue | Compilation
Authored by M.A. Gopalan , S. Vidhyalakshmi, J. Shanthi
Classification: FOR Code: MSC 2010: 11D45
Keywords: ternary quadratic, system of double equations, integer solutions.
Language: English

A new and different set of solutions is obtained for the ternary quadratic diophantine equation through representing it as a system of double equations.

               

On the Diophantine Equation

M. A. Gopalanα, S. Vidhyalakshmiσ & J. Shanthiρ

____________________________________________

  1. ABSTRACT

A new and different set of solutions is obtained for the ternary quadratic diophantine equation through representing it as a system of double equations.

Keywords: ternary quadratic, system of double equations, integer solutions.

Author α σ: Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

ρ:  Associate Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

  1. INTRODUCTION

The diophantine equation of the form where a, b, c, d are non-zero integers has been discussed by several authors [1-3]. In [4-14], integer solutions to the above equation are presented when a, b, c, d take particular numerical values. In this communication, different sets of integer solutions to the above equation are obtained when by representing it as a system of double equations involving trigonometric functions. It seems that they have not been presented earlier.

  1. METHOD OF ANALYSIS

The diophantine equation under consideration is

                                        (1)

The above equation is represented as the system of double equations as below:

System

1

2

3

Consider system: 1

Elimination of z leads to                                 (2)

 Case: 1

 Assume

                        (3)

Substituting (3) in (2), we have

and from the given system        

Case: 2

 Assume

                        (4)

Substituting (4) in (2), we have

and from the given system                

Consider system: 2

Elimination of z leads to

        

        

Case: 3

Assuming (3), the corresponding values of x, y, z are given by

                        

Case: 4

 The assumption (4) leads to

Consider system: 3

Elimination of z leads to

                (5)

Case: 5

Substituting (3) in (5), we have

and from the given system                

Case: 6

Assuming (4), the corresponding values of x, y, z are found to be

                        

It is worth to mention that the above solutions are different from the solutions presented in [15].

  1. GENERATION OF SOLUTION

If is a given solution of (1), then a second solution is found by setting

Then h is determined rationally and the second solution is represented by

The repetition of the above process leads to the generation of sequence of solutions to (1).

In conclusion, one may obtain different choices of solutions of (1) by choosing alternative forms of the system of equations or by any other method.

REFERENCES

  1. Dickson L.E., (1952), History of the theory of numbers, Vol.2, Chelsea Publishing company, New York.
  2. Batta, Bibhotibhusan and Avadhesh Narayan Singh, History of Hindu Mathematics, Asia Publishing House, Bombay (1938).
  3. Mordell L.J., (1969), Diophantine Equations, Academic Press, London.
  4. Gopalan M.A., Sangeetha V., Manju Somanath, (2013), On The Ternary Quadratic Equation, International Journal of Innovative Research in Science, Engineering and Technology, 2(6), 2008-2010.
  5. Gopalan M.A., Vidhyalakshmi S., Premalatha E., (2013), On The Homogeneous Quadratic Equation With Three Unknowns , Bulletin of Mathematics and Statistics Research, 1(1), 38-41.
  6. Gopalan M.A., Vidhyalakshmi S., Nivethitha S., (2013), On Ternary Quadratic Equation , International journal of Engineering Research Online, 1(4), 16-22.
  7. Gopalan M.A., Geetha V., (2014), Lattice Points on the Homogeneous Cone , The International Journal of Science & Technoledge, 2(7), 291-295.
  8. Gopalan M.A., Mallika S., Vidhyalakshmi S., (2014), On Ternary Diophantine Equation , Bulletin of Mathematics and Statistics Research, 2(4), 429-433.
  9. Gopalan M.A., Maheswari D., Maheswari J., (2015), On Ternary Quadratic Diophantine Equation , JP Journal of Mathematical Sciences, 13(1&2), 9-23.
  10. Gopalan M.A., Vidhyalakshmi S., Devibala S.,  Umavathy J., (2015) On the Ternary Quadratic Diophantine equation  , International Journal of Applied Research, 1(5), 234-238.
  11. Gopalan M.A., Vidhyalakshmi S., Rajalakshmi U.K., (2017) On Ternary Quadratic Diophantine equation  , IJRDO- Journal of Mathematics, Vol.3, Issue-5, 1-10.
  12. Meena K., Vidhyalakshmi S., Gopalan M.A., Aarthy Thangam S., (2014), On homogeneous ternary quadratic diophantine equation, IJESM, 3(2), 63-69.
  13. Vidhyalakshmi S., Thenmozhi S., (2017) On The Ternary Quadratic Diophantine Equation, Journal of Mathematics and Informatics, Vol 10, 11-19.
  14. Vidhyalakshmi S., Priya A., (2017), On The Non-Homogeneous Ternary Quadratic Diophantine Equation, Journal of Mathematics and Informatics, Vol 10, 49-55
  15. Dr. M.A. Gopalan, Note on the diophantine equation, Acta Ciencia Indica, Vol XXVIM, No. 2, 105-106, 2000.



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