15x^6 + 5y^6 - 20z^6  = w^2


We show diophantine equation 15x^6 + 5y^6 - 20z^6  = w^2 has infinitely many integer solutions.

15x^6 + 5y^6 - 20z^6  = w^2.................................................(1)

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

Let (a,b,c)=(15, 5, -20),(p,q)=(2, -2) the we obtain

u^2 = 19687500t^4+28125000t^3+70312500t^2+18750000t+14062500................(2)

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

Y^2 = X^3-570375X -83868750.

Rank is 2 and generator is (X,Y)=[-650,3500],[826,2926].

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

Numerical example:

[X,Y,Z,W] = [907, 1253, 367, 5258503800].