-12x^6 + 6y^6 - 48z^6 = w^2

Previously, we studied the parametric solutions of ax^6 + by^6 + cz^6 = w^2 below.
ax^6 + by^6 + cz^6 = w^2 Ⅰ
ax^6 + by^6 + cz^6 = w^2 Ⅱ


We show diophantine equation -12x^6 + 6y^6 - 48z^6 = w^2 has infinitely many integer solutions.

-12x^6 + 6y^6 - 48z^6 = w^2....................................................................(1)

We use an identity 2(t-1)^6-(2t)^6+8(t^2+2t-1)^3 = -6(t-1)^2(3t-1)^2(t+1)^2....................(2)

Let x=t-1, y=2t, z= t^2+2t-1, w= (t-1)(3t-1)(t+1) then we obtain a parametric solution of 2x^6-y^6+8z^3=-6w^2.

Furthermore, parametric solution of t^2+2t-1 =v^2 is given (t,v)=( (n^2-2n+5)/(2n^2-2), 1/2(n^2-6n+1)/((n-1)(n+1)) ).
n is arbitrary.

Thus we obtain a parametric solution -12x^6 + 6y^6 - 48z^6 = w^2 as follows.

(x,y,z,w)=( n^2+2n-7, 2n^2-4n+10, n^2-6n+1, 6(3n^2-2n+3)(n^2-6n+17)(n^2+2n-7)).

Hence equation (1) has infinitely many integer solutions.