-12x^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 -12x^6 + 6y^6 - 6z^6 = w^2 has infinitely many integer solutions.

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

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

Let x=1+t, y=2t+1, z= 2t^2-1, w=t(1+t)(3t+2) then we obtain a parametric solution of 2x^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 -12x^6 + 6y^6 - 6z^6 = w^2 as follows.

(x,y,z,w)=( 2n(n-1), 2+3n^2-4n, 2+n^2-4n,  12n(n-1)(2+n^2-2n)(5n^2-6n+2)).

Hence equation (1) has infinitely many integer solutions.