48x^6 + 24y^6 - 72z^6 = w^2


We show diophantine equation 48x^6 + 24y^6 - 72z^6 = w^2 has infinitely many integer solutions.

48x^6 + 24y^6 - 72z^6 = w^2......................................................................(1)

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

Let (a,b,c)=(48, 24, -72),(p,q)=(2, -1) the we obtain

u^2 = 577895399424t^4+1651129712640t^3+2063912140800t^2+1100753141760t+412782428160..............(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, 756].