నైరూప్య

INTEGRAL SOLUTIONS OF NONHOMOGENEOUS QUINTIC EQUATION WITH THREE UNKNOWNS

M.A.Gopalan, G.Sumathi , S.Vidhyalakshmi

The non-homogeneous quintic equation with three unknowns represented by the diophantine equation 2 2 2 5 x y xy x y 1 (k 3) z n ï�?« ï�?­ ï�?« ï�?« ï�?« ï�?½ ï�?« is analyzed for its patterns of non-zero distinct integral solutions and three different methods of integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely, polygonal numbers, Jacobsthal numbers, Jacobsthal-Lucas number,Pronic numbers, Stella octangular numbers, Octahedral numbers, Centered Polygonal numbers, Centered Pentagonal Pyramidal numbers, Centered Hexagonal Pyramidal numbers, Generalized Fibonacci and Lucas sequences, Fourth Dimensional Figurate numbers and Fifth Dimensional Figurate numbers are exhibited.