On The Homogeneous Biquadratic Equation With 5 Unknowns: x4-y4=65 (z2-w2) R2

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, Pyramidal numbers, Centered Polygonal numbers, Centered Pyramidal numbers, Thabit-ibn-Kurrah number, Star number, Carol number, woodall number, kynea number, pentatope number, stellaoctangul number, octahedral number, Mersenne number are exhibited.