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M.A.Gopalan1,V.Sangeetha2,Manju Somanath3
Professor, Department of Mathematics,Srimathi Indira Gandhi College, Trichy, India1Assistant Professor, Department of Mathematics, National College, Trichy, India2 Assistant Professor, Department of Mathematics, National College, Trichy, India3 |

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

## Keywords |

Integral solutions,Ternary quadratic |

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 |

[1] L.E.Dickson, “ History of Theory of Numbers”, Vol.2, Chelsea Publishing Company,New York, 1952. [2] L.J.Mordell, “ Diophantine Equations”, Academic Press, London, 1969. [3] M.A.Gopalan, “Note on the Diophantine equation x2 + axy + by2 = z2” , Acta Ciencia Indica, vol.XXVIM(2), pp.105-106, 2000. [4] M.A.Gopalan, “Note on the Diophantine equation x2 + xy + y2 = 3z2” , Acta Ciencia Indica, vol.XXVIM(3), pp.265-266, 2000. [5] M.A.Gopalan,R.Ganapathy and R.Srikanth, “On the Diophantine Equation z2 = Ax2 + By2”, Pure and Applied Mathematika Sciences,vol.LII(1-2), pp.15-17,2000. [6] M.A.Gopalan and R.Anbuselvi, “On ternary Quadratic Homogeneous Diophantine Equation x2 + pxy + by2 = z2”,Bulletin of Pure and Applied Sciences,vol.24E(2),pp.405-408,2005. [7] 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. [8] M.A.Gopalan,S.Vidhyalakshmi and S.Devibala, “Integral solutions of ka x2 + y2 + bxy = 4kα2z2”, Bulletin of Pure and Applied Sciences,vol.25E(2), pp.401-406,2006. [9] M.A.Gopalan,S.Vidhyalakshmi and S.Devibala ,“Integral solutions of 7x2 + 8y2 = 9z2”, Pure and AppliedMathematika Sciences,vol.LXVI(1-2),pp.83-86,2007. [10] M.A.Gopalan and S.Vidhyalakshmi, “An observation on kax2 + by2 = cz2”, Acta Cienica Indica, vol.XXXIIIM(1),pp.97-99, 2007. [11] M.A.Gopalan,Manju somanath and N.Vanitha, “Integral solutions of kxy + m x + y = z2”, Acta Cienica Indica, vol.XXXIIIM(4),pp.1287-1290, 2007. [12] M.A.Gopalan and J.Kaliga Rani, “Observation on the Diophantine Equation y2 = Dx2 + z2”,Impact J.Sci.Tech.vol.2(2),pp. 91-95, 2008. [13] M.A.Gopalan and V.Pondichelvi, “On Ternary Quadratic Equation x2 + y2 = z2 + 1”, Impact J.Sci.Tech.vol.2(2), pp.55-58, 2008. [14] M.A.Gopalan and A.Gnanam, “Pythagorean triangles and Special Polygonal numbers”, International Journal of Mathematical Sciences,vol.9(1-2),pp. 211-215, Jan-June 2010. [15] M.A.Gopalan and A.Vijayasankar, “Observations on a Pythagorean Problems”,Acta Cienica Indica,vol.XXXVIM (4), pp.517- 520, 2010. [16] M.A.Gopalan and V.Pondichelvi, “Integral solutions of Ternary Quadratic Equation Z X − Y = 4XY”, Impact J.Sci.Tech.vol.5(1),pp.1-6, 2011. [17] M.A.Gopalan and J.Kaliga Rani, “On Ternary Quadratic Equation X2 + Y2 = Z2 + 8”, Impact J.Sci.Tech.vol.5(1),pp.39-43,2011. [18] M.A.Gopalan,S.Vidhyalakshmi and T.R.Usharani, “Integral Points on the Homogeneous Cone 2z2 + 4xy + 8x − 4z + 2 = 0”,Global Journal of Mathematics and Mathematical Sciences, vol.2 (1),pp.61-67,2012. [19] 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. [20] 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. [21] 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. [22] 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. [23] 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. |