-48x^6 + 6y^6 - 6z^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 -48x^6 + 6y^6 - 6z^6 = w^2 has infinitely many integer solutions.

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

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

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

Furthermore, parametric solution of 2t^2-1 =v^2 is given (t,v)=((2+n^2-2n)/(-2+n^2), -(2+n^2-4n)/(-2+n^2)).
n is arbitrary.

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

(x,y,z,w)=(2n-4, n^2-4n+6, n^2-4n+2, 12*(n-2)(n^2-4n+6)(n^2-4n+2) ).

Hence equation (1) has infinitely many integer solutions.