On the Heptic Non-Homogeneous Equation with Four Unknowns xy(x+y) + zw6 =0

The heptic non-homogeneous equation with four unknowns represented by the diophantine equation ( ) 2 0 6 xy x + y + zw = is analyzed for its patterns of non-zero distinct integral solutions and two different methods of integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely, Pyramidal numbers, Pronic numbers, Star numbers, Gnomonic numbers, Centered Polygonal numbers, Centered Hexagonal Pyramidal numbers, fourth dimensional figurate numbers are exhibited.