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The homogeneous cubic equation with four unknowns represented by is analyzed for its patterns of non zero distinct integral solutions.Three different patterns of solutions are presented. A few interesting relations between the solutions and special numbers ,namely, polygonal numbers,pyramidal numbers,centered pyramidal numbers,star numbers, nasty numbers,dodecahedral number,rhombic dodecahedral number and prism numbers are exhibited.

LATTICE POINTS ON THE HOMOGENEOUS CUBIC EQUATION WITH FOUR UNKNOWNS x2 - xy + y2 + 4w2 = 8z3