International Research Journal of Engineering and Technology (IRJET) Volume: 04 Issue: 03 | Mar -2017
e-ISSN: 2395 -0056
www.irjet.net
p-ISSN: 2395-0072
OBSERVATIONS ON TERNARY QUADRATIC EQUATION
z 2 82 x 2 y 2
G. Janaki1 and S. Vidhya2 1,2Assistant
Professor, Department of Mathematics, Cauvery College for Women, Trichy-18, Tamilnadu, India.
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Abstract-The Quadratic Diophantine Equation with three unknowns z2 = 82x2 + y2 is analyzed for its non-zero distinct integer solutions. A few interesting relations among the solutions and some special polygonal, Octahedral and Four dimensional figurate numbers are presented. Keywords Quadratic equation with three unknowns, Integral solutions, Polygonal Numbers, Four Dimensional Figurate Numbers, Octahedral number.
1. Introduction Ternary quadratic equations are rich in variety. For an extensive review of sizable literature and various problems one may refer [1-7]. In [8], the ternary quadratic Diophantine equation of the form kxy m( x y ) z has been studied for its non-trivial integral solutions. In [9,10], A special Pythagorean triangle problem have been discussed for its integral solutions. In [11], two parametric non-trivial integral solutions of the ternary quadratic homogeneous Diophantine equation X 2 PXY Y 2 Z 2 , where P is a non-zero constant have been presented. In [12], the ternary quadratic homogeneous 2 2 2 2 equation k ( x y ) bxy 4k z , (k, ,b 0) has been studied for its non-trivial integral solutions. In [13], the 2
ternary quadratic Diophantine equation of the form ( x y)( x z) y angles and their parametric representations are obtained.
2
0, is analyzed for its integral solutions at different
In this communication, we consider yet another interesting ternary quadratic equation z2 = 82x2 + y2 and obtain different patterns of non-trivial integral solutions. Also, a few interesting relations among the solutions and special polygonal, Octahedral and Four dimensional numbers are presented.
2. Notations Denoting the ranks of some special polygonal, Octahedral and Four dimensional figurate numbers.
n 1m 2 Tm,n n 1 = Polygonal Number with rank n and sides m . 2
On
( 2n 3 n) = Octahedral Number of rank n. 3
4 DFn
n 2 (n 2 1) = Four dimensional figurate Number whose generating polygon is a square. 12
3.Method of Analysis The ternary quadratic Diophantine equation to be solved for its non-zero integral solution is
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