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OBSERVATIONS ON THE NONHOMOGENEOUS SEXTIC EQUATION WITH FOUR UNKNOWNS

S.Vidhyalakshmi1, S.Mallika2, M.A.Gopalan3
Assistant Professor of Mathematics , SIGC, Trichy-2,Tamilnadu,India 1
Lecturer of Mathematics , SIGC, Trichy-2,Tamilnadu,India 2
Assistant Professor of Mathematics , SIGC, Trichy-2,Tamilnadu,India3
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Abstract

The sextic non-homogeneous equation with four unknowns represented by the Diophantine equation x y k z w 3 3 2 5   2(  3) is analyzed for its patterns of non-zero distinct integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely polygonal numbers, Pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas number, Pronic numbers, Star numbers are exhibited.

Keywords

Quintic equation with five unknowns, Integral solutions, centered polygonal numbers, centered pyramidal numbers.
Mathematics subject classification number: 11D41.

NOTATIONS

- Polygonal number of rank n with size m.
- Pyramidal number of rank n with size m.
- Pentatope number of rank n
-Stella octangular number of rank n
-Star number of rank n
- Pronic number of rank n.
Jacobsthal number of rank n.
- Jacobsthal lucas number of rank n.
- Keynea number.
- Four dimensional figurative number of rank n
whose generating polygon is a triangle.
-Five dimensional figurative number of rank n whose generating polygon is a triangle.
- Centered polygonal number of rank n with size m.

I.INTRODUCTION

The theory of Diophantine equations offers a rich variety of fascinating problems. In particular quintic equations homogeneous or non-homogeneous have aroused the interest of numerous mathematicians since antiquity [1,2,3].For illustration ,one may refer [4-10],for quintic equations with three ,four and five unknowns. This paper concerns with the problem of determining integral solutions of the non-homogeneous quintic equation with five unknowns given by . A few relations between the solutions and the special numbers are presented..

II.METHOD OF ANALYSIS

The Diophantine equation representing the quintic with five unknowns under consideration is
(1)
Introducing the transformations
(2)
where  is a distinct positive distinct integer in (1),we get
(3)
Assume (4)
Substituting (4) in (3) and employing the method of factorization define
(5)
equating real and imaginary parts, we get
Thus, in view of (2), the non-zero distinct integral solutions of (1) are given by
For simplicity and clear understanding we present below the integer solutions and the corresponding properties for α 0 and α1.
A. Case:1
Let α 0 The non-zero distinct integer solutions of (1) are found to be
B.Properties
1) Remark : It is worth to note that when α 0,we have another pattern of solution which is illustrated below.
For this case α 0 ,(3) reduces to
(6)
Following the analysis presented above ,the values of u and v are given by
Hence, the non-zero distinct integral solutions of (1) are given by,
The above values of x, y, z and w are different from that of in case (1) presented above.
C.Properties:
D.Case:2
Let α1 .After performing a few calculations as in case (1) the non-zero distinct integer solutions are obtained as
E.Properties:

III.CONCLUSION

In addition to the above patterns of solutions, there are other forms of integer solutions to (1). For illustration, whenα 0, the equation (6) is written as
(7)
Write 1 as,
(8)
or
(9)
Using (4) and (8) in (7) and employing the method of factorization, define,
Equating the real and imaginary parts, the values of u and v are obtained.Substuting these values of u and v in (2) and choosing a and b suitably, many different integer solutions to (1) are obtained. Similar process is carried out by considering (4) and (9).
To conclude one may search for other choices of solutions to (1) along with the corresponding properties.

References

[1]. L.E.Dickson, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York (1952).
[2] L.J.Mordell, Diophantine equations, Academic Press, London(1969).
[3] Carmichael ,R.D.,The theory of numbers and Diophantine Analysis,Dover Publications, New York (1959)
[4] M.A.Gopalan & A.Vijayashankar, An Interesting Diophantine problem , Advances in Mathematics, Scientific Developments and Engineering Application, Narosa Publishing House, Pp 1-6, 2010.
[5] M.A.Gopalan & A.Vijayashankar, Integral solutions of ternary quintic Diophantine equation ,International Journal of Mathematical Sciences 19(1-2), 165-169,(jan-june 2010)
[6] M.A.Gopalan,G.Sumathi & S.Vidhyalakshmi, Integral solutions of non-homogeneous ternary quintic equation in terms of pells sequence,accepted for Publication in JAMS(Research India Publication)
[7]. S.Vidhyalakshmi, K.Lakshmi and M.A.Gopalan, Observations on the homogeneousquintic equation with four unknowns ,accepted for Publication in International Journal of Multidisciplinary Research Academy (IJMRA).
[8] M.A.Gopalan & A.Vijayashankar, Integral solutions of non-homogeneous quintic equation with five unknowns
, Bessel J.Math.,1(1),23-30,2011.
[9]. M.A.Gopalan & A.Vijayashankar, solutions of quintic equation with five unknowns ,Accepted for Publication in International Review of Pure and Applied Mathematics.
[10] M.A.Gopalan, G. Sumathi & S.Vidhyalakshmi, On the non-homogenous quintic equation with five unknowns
,accepted for Publication in International Journal of Multidisciplinary Research Academy (IJMRA).