16x^6 + 2y^6 - 18z^6  = w^2


We show diophantine equation 16x^6 + 2y^6 - 18z^6  = w^2 has infinitely many integer solutions.

16x^6 + 2y^6 - 18z^6  = w^2.................................................(1)

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

Let (a,b,c)=(16, 2, -18),(p,q)=(2, -7) the we obtain

u^2 = 15123456t^4-12718080t^3+4838400t^2-737280t+138240.....................(2)

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

Y^2 = X^3-79935X -3993850.

Rank is 2 and generator is (X,Y)=[1178779/81,-1279576718/729],[-10254410/47961,-19146701350/10503459].

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

Numerical example:

[X,Y,Z,W] = [10, 17, 7, 7884].