[Gopalan et al., Vol.1 (Iss.1): April, 2015] ISSN: 2454-1907 IJETMR International Journal of Engineering Technologies and Management Research A knowledge Repository Http://www.ijetmr.com©International Journal of Engineering Technologies and Management Research [14-22] ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION 6z 2 = 6x 2 – 5y 2 M. A. Gopalan *1 , S. Nandhini 2 , J. Shanthi 3 *1 Professor, Department of Mathematics, SIGC, Trichy-620002, Tamil Nadu, INDIA 2 M.Phil Scholar, Department of Mathematics, SIGC, Trichy-620002, Tamil Nadu, INDIA 3 Lecturer, Department of Mathematics, SIGC, Trichy-620002, Tamil Nadu, INDIA Abstract: The ternary homogeneous quadratic equation given by 2 2 2 5y - 6x 6z  representing a cone is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and special polygonal and pyramided numbers are presented. Also, given a solution, formulas for generating a sequence of solutions based on the given solutions are presented. Keywords: Ternary quadratic, integer solutions, figurate numbers, homogeneous quadratic, polygonal number, pyramidal numbers. Notations Used: 1. Polygonal number of rank ‘n’ with sides m           2 ) 2 m )( 1 n ( 1 n t n , m m=3, 4, 5…, 10 … 2. Stella octangular number of rank ‘n’ ) 1 n 2 ( n SO 2 n   Cite This Article: M. A. Gopalan, S. Nandhini, and J. Shanthi, “On the Ternary Quadratic Diophantine Equation 6z 2 = 6x 2 – 5y 2 .” International Journal of Engineering Technology and Management Research, Vol. 1, No. 1(2015): 14-22. 1. INTRODUCTION The Diophantine equations offer an unlimited field for research due to their variety [1-3].In particular, one may refer [4-24] for quadratic equations with three unknowns. This communication concerns with yet another interesting equation 2 2 2 5y - 6x 6z  representing non-homogeneous quadratic equation with three unknowns for determining its infinitely many non-zero integral points. Also, a few interesting relations among the solutions are presented.