16x^6 + 9y^6 - 25z^6  = w^2


We show diophantine equation 16x^6 + 9y^6 - 25z^6  = w^2 has infinitely many integer solutions.

16x^6 + 9y^6 - 25z^6  = w^2.................................................(1)

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

Let (a,b,c)=(16, 9, -25),(p,q)=(2, -7/9) the we obtain

u^2 = 531968400t^4+1544702400t^3+1867698000t^2+1049760000t+354294000........(2)

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

Y^2 = X^3-3470415X -1372046410.

Rank is 1 and generator is (X,Y)=[-3243320666/2082249,-47937055961998/3004685307].

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

Numerical example:

[X,Y,Z,W] = [34, 41, 7, 259740].