30x^6 + 20y^6 - 50z^6 = w^2


We show diophantine equation 30x^6 + 20y^6 - 50z^6 = w^2 has infinitely many integer solutions.

30x^6 + 20y^6 - 50z^6 = w^2..................................................................(1)

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

Let (a,b,c)=(30, 20, -50),(p,q)=(2, -1/2) the we obtain

u^2 = 119700000000t^4+349200000000t^3+414000000000t^2+240000000000t+72000000000..............(2)

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

Y^2 = X^3-48735X -2337930.

Rank is 1 and generator is (X,Y)=[-499705276794/5434196089,-468191042430719994/400592633092813].

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

Numerical example:

[X,Y,Z,W] = [221, 389, 23, 269804220].