30x^6 + 6y^6 - 36z^6  = w^2


We show diophantine equation 30x^6 + 6y^6 - 36z^6  = w^2 has infinitely many integer solutions.

30x^6 + 6y^6 - 36z^6  = w^2.................................................(1)

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

Let (a,b,c)=(30, 6, -36),(p,q)=(2, -4) the we obtain

u^2 = 1234517760*t^4-1461265920*t^3+1385683200*t^2-167961600*t+125971200..............(2)

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

Y^2 = X^3-360375X -39488750.

Rank is 1 and generator is (X,Y)=[-308772066566/2396200401,257228766926568998/117296405829351].

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

Numerical example:

[X,Y,Z,W] = [275, 139, 206, 101328570].