28x^6 + 14y^6 - 42z^6 = w^2


We show diophantine equation 28x^6 + 14y^6 - 42z^6 = w^2 has infinitely many integer solutions.

28x^6 + 14y^6 - 42z^6 = w^2.............................................................(1)

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

Let (a,b,c)=(28, 14, -42),(p,q)=(2, -1) the we obtain

u^2 = 13282101504t^4+37948861440t^3+47436076800t^2+25299240960t+9487215360..............(2)

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

Y^2 = X^3-324135X -38433150.

Rank is 2 and generator is (X,Y)=[714,9702],[-150,2610].

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

Numerical example:

[X,Y,Z,W] = [2, 11, 5, 4914].