International Research Journal of Engineering and Technology (IRJET) Volume: 04 Issue: 3 | Mar -2017
www.irjet.net
e-ISSN: 2395 -0056 p-ISSN: 2395-0072
INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION 3( x 2 y 2 ) 4 xy 2( x y 1) 972z 3 G.Janaki1, C.Saranya2,* Assistant Professors, Department of Mathematics, Cauvery College for Women, Tiruchirappalli, Tamil Nadu, India. ---------------------------------------------------------------------------------------------------------------------------------------------------------Abstract:The non-homogeneous cubic equation with three unknowns represented by the Diophantine equation 3( x 2 y 2 ) 4 xy 2( x y 1) 972 z 3 is analyzed for its patterns of non – zero integral solutions. A few interesting properties among the solutions and some special polygonal numbers are presented. Keywords: Cubic Equation with Three Unknowns, Integral solutions Introduction: Mathematics is the language of patterns and relationships and is used to describe anything that can be quantified. Diophantine equations has been matter of interest to various mathematicians. The problem of finding all integer solutions of a diophantine equation with three or more variables and equations of degree at least three, in general presents a good deal of difficulties. In [1-3], theory of numbers were discussed. In [4,5], a special Pythagorean triangle problem have been discussed for its integral solutions. In [6-10], higher order equations are considered for integral solutions. In this communication, the non-homogeneous cubic equation with three unknowns represented by the equation 3( x 2 y 2 ) 4 xy 2( x y 1) 972 z 3 is considered and in particular a few interesting relations among the solutions are presented. Notations: Obl n = Oblong number of rank ‘n’.
Tm, n = Polygonal number of rank ‘n’ with sides ‘m’.
CS n = Centered Square number of rank ‘n’. SOn = Stella octangula number of rank ‘n’. On = Octahedral number of rank ‘n’. Gno n = Gnomonic number of rank ‘n’. Starn = Star number of rank ‘n’. TOn = Truncated octahedral number of rank ‘n’. Pnm = Pyramidal number of rank ‘n’ with sides ‘m’.
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