48x^6 + 12y^6 - 60z^6 = w^2


We show diophantine equation 48x^6 + 12y^6 - 60z^6 = w^2 has infinitely many integer solutions.

48x^6 + 12y^6 - 60z^6 = w^2.................................................(1)

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

Let (a,b,c)=(48, 12, -60),(p,q)=(2, -3) the we obtain

u^2 = 35115171840t^4-25798901760t^3+75246796800t^2+10749542400..............(2)

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

Y^2 = X^3-1488375X -339558750.

Rank is 1 and generator is (X,Y)=[-1050 , -8100].

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

Numerical example:

[X,Y,Z,W] = [251, 971, 395, 3137158080].