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# ON THE TERNARY QUADRATIC EQUATION

 M.A.Gopalan1,V.Sangeetha2,Manju Somanath3 Professor, Department of Mathematics,Srimathi Indira Gandhi College, Trichy, India1 Assistant Professor, Department of Mathematics, National College, Trichy, India2 Assistant Professor, Department of Mathematics, National College, Trichy, India3 Related article at Pubmed, Scholar Google

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## Abstract

The ternary quadratic Diophantine equation 5 x2 + y2 − 9xy = 19z2 is analyzed for its patterns of non zero distinct integral solutions

### Keywords

MSC 2000 subject classification number 11D09

### INTRODUCTION

The Ternary Quadratic Diophantine Equation offers an unlimited field for research because of their variety [1, 2]. For an extensive review of various problems, one may refer [3-23]. This communication concerns with yet another interesting Ternary Quadratic Equation for determining its infinitely many non-zero integral solutions.

### II.METHOD OF ANALYSIS

The ternary quadratic equation to be solved in integers is It is noted that (1) can be satisfied by the following triples of integers (1141a,379a,419a),(-2294A,1895A,2095A),(151a,a,77a),(746A,5A,385A),(305T,94T,115T), (-679A,470A,575A),(346a,91a,137a),(-911A,455A,685A),(1469A,355A,595A), (-166a,71a,119a),(481a,70a,215a),(-71T,14T,43T).
Now,introducing the linear transformations
�� = �� + �� ; �� = �� − �� (2)  which is equivalent to the system of equations  It is observed that, by rewriting (4) suitably, one may arrive at the following three patterns of solutions to (1). However , we have another solution pattern which is illustrated below: ### III.CONCLUSION

To conclude, one may search for other patterns of solutions to the equation under consideration.

### References

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 M.A.Gopalan,S.Vidhyalakshmi and A.Krishnamoorthy, “Integral solutions of Ternary Quadratic ax2 + by2 = c(a + b)z2” Bulletin of Pure and Applied Sciences,vol.24E(2),pp.443-446,2005.
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 M.A.Gopalan,S.Vidhyalakshmi and T.R.Usharani, “Integral Points on the Homogeneous Cone 2z2 + 4xy +
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 M.A.Gopalan,S.Vidhyalakshmi,T.R.Usharani and S.Mallika “Integral Points on the Homogeneous Cone 6z2 + 3y2 − 2x2 = 0”, Impact J.Sci.Tech.,vol.6(1),pp.7-13, 2012.
 M.A.Gopalan,S.Vidhyalakshmi and G.Sumathi, “Lattice Points on The Hyperboloid of One Sheet 4z2 = 2x2 + 3y2 − 4”,Diophantus J.Math.vol.1(2),pp. 109-115,2012.
 M.A.Gopalan,S.Vidhyalakshmi and G.Sumathi, “Lattice Points on The Elliptic Paraboloid 9x2 + 4y2 = z”,Advances in Theoretical and Applied Mathematics,vol.7(4),pp.379-385,2012.
 M.A.Gopalan,S.Vidhyalakshmi and K.Lakshmi, “Integral Points on The Hyperboloid of Two Sheets 3y2 = 7x2 − z2 + 21”, Diophantus J.Math.vol.1(2),pp.99-107,2012.
 M.A.Gopalan,V.Sangeetha and Manju Somanath, “Integral Points on the homogeneous cone z2 = 5x2 + 11y2”,Discovery Science,vol.3(7),pp.5-8, 2013.