ISSN ONLINE(2319-8753)PRINT(2347-6710)
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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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 |
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whose generating polygon is a triangle. |
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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 ![]() |
II.METHOD OF ANALYSIS |
The Diophantine equation representing the quintic with five unknowns under consideration is |
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Introducing the transformations |
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where ïÃÂá is a distinct positive distinct integer in (1),we get |
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Assume ![]() |
Substituting (4) in (3) and employing the method of factorization define |
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equating real and imaginary parts, we get |
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Thus, in view of (2), the non-zero distinct integral solutions of (1) are given by |
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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 |
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B.Properties |
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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 |
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Following the analysis presented above ,the values of u and v are given by |
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Hence, the non-zero distinct integral solutions of (1) are given by, |
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The above values of x, y, z and w are different from that of in case (1) presented above. |
C.Properties: |
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D.Case:2 |
Let α1 .After performing a few calculations as in case (1) the non-zero distinct integer solutions are obtained as |
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E.Properties: |
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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 |
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Write 1 as, |
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or |
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Using (4) and (8) in (7) and employing the method of factorization, define, |
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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 |
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