On The Homogeneous Biquadratic Equation with 5 Unknowns

The Homogenous biquadratic equation with five unknowns given by is considered and analyzed for finding its non zero distinct integral solutions. Introducing the linear transformations x = u + v, y = u v, z = 2uv +1, w =2uv-1 and employing the method of factorization different patterns of non zero distinct integer solutions of the equation under the above equation are obtained. A few interesting relations between the integral solutions and the special numbers namely Polygonal numbers, Star number, Pronic number and Gnomonic number are exhibited.