Five different methods of the non-zero integral solutions of the homogeneous biquadratic Diophantine equation with five unknowns 7(x 2 + y 2)-13xy = 31z 4 are determined. Introducing the linear transformations x = u + v, y = u-v, u  v  0 in 7(x 2 + y 2)-13xy = 31z 4 , it reduces to u 2 +27v 2 = 31z 4. We are solved the above equation through various choices and the different methods of solutions which are satisfied it. Some interesting relations among the special numbers and the solutions are

more »

... exposed Notations used T m,n : Polygonal number of rank n with sides m. m n p : Pyramidal number of rank m with side n G n : Gnomonic number of rank n 4,3 r f : Fourth dimensional figurate number of rank r, whose generating polygon is a Triangle 4,4 r f : Fourth dimensional figurate number of rank r, whose generating polygon is a Square 4,5 r f : Fourth dimensional figurate number of rank r, whose generating polygon is a Pentagon 4,6 r f : Fourth dimensional figurate number of rank r, whose generating polygon is a Hexagon 4,7 r f : Fourth dimensional figurate number of rank r, whose generating polygon is a Heptagon 4,8 r f : Fourth dimensional figurate number of rank r, whose generating polygon is a Octagon.