On Homogeneous Quinary Quadratic Diophantine Equation
S. Vidhyalakshmi ^{1}, M.A. Gopalan ^{2}
^{1}^{ }Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University, Trichy620 002, Tamil Nadu, India
^{2} Professor, Department of Mathematics, Shrimati Indira Gandhi
College, Affiliated to Bharathidasan University, Trichy620 002, Tamil Nadu,
India

ABSTRACT 

The
homogeneous quadratic Diophantine equation with five unknowns given by is analyzed for determining its nonzero distinct integer solution through employing linear transformations. 

Received 14 April 2022 Accepted 13 May 2022 Published 09 June 2022 Corresponding Author M.A.
Gopalan, mayilgopalan@gmail.com DOI 10.29121/granthaalayah.v10.i5.2022.4623 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Homogeneous Quadratic, Quadratic with
Five Unknowns, Integer Solutions 
1. INTRODUCTION
The theory of Diophantine equations offers a rich variety of fascinating problems. In particular, homogeneous, or nonhomogeneous quadratic Diophantine equations with two or more variables have been an interest to mathematicians since antiquity Dickson (1971), Mordell (1969), Andre (1984), Datta and Singh (1938). In this context, one may refer Gopalan and Srividhya (2012), Gopalan et al. (2013), Vijayasankar et al. (2017), Vidhyalakshmi et al. (2018), Adiga (2020) for different choices of quadratic Diophantine equations with four unknowns. In Anbuselvi and Rani (2017), Anbuselvi and Rani (2018), Gopalan et al. (2013) the quadratic Diophantine equation with five unknowns are analysed for obtaining their nonzero distinct integer solutions.
This motivated me for finding integer solutions to other choices of quadratic equations with five unknowns. This paper deals with the problem of determining nonzero distinct integer solutions to the quadratic Diophantine equation with five unknowns given by
2. METHOD OF ANALYSIS
The seconddegree Diophantine equation with five unknowns to be solved is
The process of obtaining different sets of nonzero distinct integer solutions
To Equation 1 is exhibited below:
Set 1:
The substitution of the linear transformations
in Equation 1 leads to the Pythagorean equation
Equation 3
which is satisfied by
In view of Equation 2, one has
Thus, Equation 4 and Equation 5 represent the integer solutions to Equation 1
Set 2:
Introducing the linear transformations
in Equation 1 it simplifies to the Pythagorean equation
Equation 7
whose solutions may be taken as
Equation 8
In view of Equation 6 the integer solutions to Equation 1 are given by
Set 3:
Taking
in Equation 1 it reduces to
Treating Equation 10 as a quadratic in z and solving for z, it is seen that Equation 10 is satisfied by
In view of Equation 9 it is seen that the corresponding values of x, y, w satisfying Equation 1 are
Set 4:
Taking
in Equation 1 it reduces to
Treating Equation 12 as a quadratic in z and solving for z, it is seen that Equation 12 is satisfied by
In view of Equation 11 it is seen that the corresponding values of satisfying Equation 1 are
3. CONCLUSION
In this paper, an attempt has been made to obtain nonzero distinct integer solutions to the quadratic Diophantine equation with five unknowns given by The readers of this paper may search for finding integer solutions to other choices of quadratic Diophantine equations with five or more unknowns.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
REFERENCES
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