36x^6 + 12y^6 - 48z^6 = w^2


We show diophantine equation 36x^6 + 12y^6 - 48z^6 = w^2 has infinitely many integer solutions.

36x^6 + 12y^6 - 48z^6 = w^2..........................................................(1)

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

Let (a,b,c)=(36, 12, -48),(p,q)=(2, -2) the we obtain

u^2 = 9029615616t^4+12899450880t^3+32248627200t^2+8599633920t+6449725440..............(2)

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

Y^2 = X^3-2535X -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, 17556].