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

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The quintic Diophantine equation with five unknowns 4 4 2 2 2 2 3 x Ã¯ÂÂ y Ã¯ÂÂ½ 2(k Ã¯ÂÂ« s )(z Ã¯ÂÂw )p is analyzed for its infinitely many non-zero distinct integral solutions. A few interesting relations between the solutions and special numbers namely, centered polygonal numbers, centered pyramidal numbers, jacobsthal numbers, Lucas numbers and Keynea numbers are presented.

## 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). |