10x^6 + 2y^6 - 12z^6  = w^2


We show diophantine equation 10x^6 + 2y^6 - 12z^6  = w^2 has infinitely many integer solutions.

10x^6 + 2y^6 - 12z^6  = w^2.................................................(1)

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

Let (a,b,c)=(10, 2, -12),(p,q)=(2, -4) the we obtain

u^2 = 564480t^4-668160t^3+633600t^2-76800t+57600............................(2)

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

Y^2 = -3243375X -1066196250.

Rank is 3 and generator is (X,Y)=[ -1550,15400],[11650,1241900],[24830950/12321,-33015093650/1367631].

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

Numerical example:

[X,Y,Z,W] = [55, 487, 127, 163190160].