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INTEGRAL SOLUTIONS OF NONHOMOGENEOUS QUINTIC EQUATION WITH THREE UNKNOWNS

M.A.Gopalan1 , G.Sumathi2 , S.Vidhyalakshmi3
Professor, PG and Research, Department of Mathematics, Shrimati Indira Gandhi, College,Trichy-620002,Tamilnadu,India1
Lecturer, PG and Research, Department of Mathematics, Shrimati Indira Gandhi College , Trichy-620002,Tamilnadu,India2
Professor, PG and Research, Department of Mathematics , Shrimati Indira Gandhi College, Trichy-620002,Tamilnadu,India3
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Abstract

The non-homogeneous quintic equation with three unknowns represented by the diophantine equation 2 2 2 5 x y xy x y 1 (k 3) z n        is analyzed for its patterns of non-zero distinct integral solutions and three different methods of integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely, polygonal numbers, Jacobsthal numbers, Jacobsthal-Lucas number,Pronic numbers, Stella octangular numbers, Octahedral numbers, Centered Polygonal numbers, Centered Pentagonal Pyramidal numbers, Centered Hexagonal Pyramidal numbers, Generalized Fibonacci and Lucas sequences, Fourth Dimensional Figurate numbers and Fifth Dimensional Figurate numbers are exhibited.

Keywords

Integral solutions, Generalized Fibonacci and Lucas sequences, Quintic non-homogeneous equation with three unknowns
M.Sc 2000 mathematics subject classification: 11D25

NOTATIONS

tm,n : Polygonal number of rank n with size m
Prn : Pronic number of rank n
Son : Stella octangular number of rank n
jn : Jacobsthal lucas number of rank n
J n : Jacobsthal number of rank n
GFn : Generalized Fibonacci sequence number of rank n
GLn : Generalized Lucas sequence number of rank n
Ctm,n : Centered Polygonal number of rank n with size m
Cf3,n,5 : Centered Pentagonal Pyramidal number of rank n
Cf3,n,6 : Centered Hexogonal Pyramidal number of rank n
f4,n,3 : Fourth Dimensional Figurate Traingular number of rank n
f4,n,4 : Fourth Dimensional Figurate Square number of rank n
f4,n,6 : Fourth Dimensional Figurate Hexogonal number of rank n
f5,n,3 : Fifth Dimensional Figurate Traingular number of rank n
f5,n,7 : Fifth Dimensional Figurate Heptagonal number of rank n

INTRODUCTION

The theory of diophantine equations offers a rich variety of fascinating problems. In particular,quintic equations, homogeneous and non-homogeneous have aroused the interest of numerous mathematicians since antiquity[1-3].For illustration, one may refer [4-5] for quintic equation with three unknowns and [6-8] for quintic equation with five unknowns. This communication concerns with an interesting non-homogeneous ternary quintic equation with three unknowns represented by
image
for determining its infinitely many non-zero integral points. Three different methods are illustrated. In method 1, the solutions are obtained through the method of factorization. In method 2, the binomial expansion is introduced to obtain the integral solutions. In method 3, the integral solutions are expressed in terms of Generalized Fibonacci and Lucas sequences along with a few properties in terms of the above integer sequences Also, a few interesting relations among the solutions are presented.

II.METHOD OF ANALYSIS

The Diophantine equation representing a non-homogeneous quintic equation with three unknowns is
image (1)
Introducing the linear transformations
image (2)
in (1), it leads to
image (3)
The above equation (3) is solved through three different methods and thus, one obtains three distinct sets of solutions to (1).
A. Method:1
Let image (4)
Substituting (4) in (3) and using the method of factorization,define
image (5)
image(6)
Equating real and imaginary parts in (5) we get
image
image
Substituting the values of u and v in (2), the corresponding values of x, y, z are represented by
image
image
image
A few numerical examples are given below:
Table: Numerical Examples:
B. Method 2:
Using the binomial expansion of image in (5) and equating real and imaginary parts, we have
image
image
Where
image
In view of (2) and (7) the corresponding integer solution to (1) is obtained as
image
image
C. Method 3:
Taking n = 0 and u +1=U in (3), we have,
image(8)
Substituting (4) in (8), we get
image(9)
whose solution is given by
image
image
Again taking n =1 in (3), we have
image(10)
whose solution is represented by
image
The general form of integral solutions to (1) is given by
image
Where
image
image
Thus, in view of (2), the following of integers xs , ys interms of Generalized Lucas and fiboanacci sequence satisfy (1) are as follows:
image
image
The above values of xs , ys satisfy the following recurrence relations respectively
image
Properties
image
image
image
image
image
image
is a quintic integer.
7.Each of the following is a nasty number.
image
image
image
image
image

III. CONCLUSION

To conclude, one may search for other pattern of solutions and their corresponding properties.

Tables at a glance

Table icon Table icon Table icon Table icon Table icon
Table 1 Table 2 Table 3 Table 4 Table 5
 

Figures at a glance

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5
Figure 1 Figure 2 Figure 3 Figure 4 Figure 5
 

References

1. L.E.Dickson, History of Theory of Numbers, Chelsea Publishing company, Vol.11, 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 Dio.problem image, Advances in Mathematics, Scientific Developments and Engineering Application, Narosa Publishing House, 2010, Pp 1-6.
5 .M.A.Gopalan & A.Vijayashankar, Integrated solutions of ternary quintic Diophantine equation image,International Journal of Mathematical Sciences 19(1-2),(jan-june 2010), 165-169.
6. M.A.Gopalan & A.Vijayashankar, Integrated solutions of non-homogeneous quintic equation with five unknowns
image , Bessel J.Math.,1(1), 2011,23-30.
7. M.A.Gopalan & A.Vijayashankar, solutions of quintic equation with five unknowns image ,Accepted for Publication in International Review of Pure and Applied Mathematics.
8. M.A.Gopalan G.Sumathi and S.Vidhyalakshmi, On the non-homogeneous quintic equation with five
unknowns image ,Accepted for Publication in rijmie journal.