6x^6 + 4y^6 - 10z^6 = w^2


Solution method of ax^6 + by^6 + cz^6 = w^2 is given below.
ax^6 + by^6 + cz^6 = w^2 Ⅴ

We show diophantine equation 6x^6 + 4y^6 - 10z^6 = w^2 has infinitely many integer solutions.

6x^6 + 4y^6 - 10z^6 = w^2.....................................................(1)

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

Let (a,b,c)=(6, 4, -10),(p,q)=(2, -1/2) the we obtain

u^2 =1532160t^4+4469760t^3+5299200t^2+3072000t+921600.........................(2)

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

Y^2 = -1218375X -292241250.

Rank is 1 and generator is (X,Y)=[-950 , 2800].

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

Numerical example:

[X,Y,Z,W] = [8777, 15497, 11465, 5952830002560].