33x^6 + 3y^6 - 36z^6  = w^2


We show diophantine equation 33x^6 + 3y^6 - 36z^6  = w^2 has infinitely many integer solutions.

33x^6 + 3y^6 - 36z^6  = w^2.................................................(1)

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

Let (a,b,c)=(33, 3, -36),(p,q)=(2, -10) the we obtain

u^2 = 2188513404t^4-1307738520t^3+333430020t^2-40415760t+4330260............(2)

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

Y^2 = X^3-32105535X -31777425350.

Rank is 1 and generator is (X,Y)=[-22804110/22201 , 35712600650/3307949].

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

Numerical example:

[X,Y,Z,W] = [17, 7, 15, 19668].