12x^6 + 6y^6 - 18z^6  = w^2


We show diophantine equation 12x^6 + 6y^6 - 18z^6  = w^2 has infinitely many integer solutions.

12x^6 + 6y^6 - 18z^6  = w^2.................................................(1)

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

Let (a,b,c)=(12, 6, -18),(p,q)=(2, -1) the we obtain

u^2 = 35271936t^4+100776960t^3+125971200t^2+67184640t+25194240..............(2)

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

Y^2 = X^3-735X -4150.

Rank is 2 and generator is (X,Y)=[50,290],[-9890/441,-300700/9261].

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

Numerical example:

[X,Y,Z,W] = [4, 5, 1, 378].