9x^6 + 3y^6 - 12z^6   = w^2


We show diophantine equation 9x^6 + 3y^6 - 12z^6   = w^2 has infinitely many integer solutions.

9x^6 + 3y^6 - 12z^6   = w^2.................................................(1)

For details,ax^6 + by^6 + cz^6 = w^2 Ⅴ

Let (a,b,c)=(9, 3, -12),(p,q)=(2, -2) the we obtain

u^2 = 551124t^4+787320t^3+1968300t^2+524880t+393660.........................(2)

This quartic equation is birationally equivalent to an elliptic curve below.

Y^2 = -2535X -24850.

Rank is 2 and generator is (X,Y)=[1994 , 89012 ],[-61010/1521, -6577550/59319].

Hence we can obtain infinitely many integer solutions for equation (1).

Numerical example:

[X,Y,Z,W] = [-13, 15, -6, 8778].