25x^6 + 20y^6 - 45z^6 = w^2


We show diophantine equation 25x^6 + 20y^6 - 45z^6 = w^2 has infinitely many integer solutions.

25x^6 + 20y^6 - 45z^6 = w^2............................................................(1)

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

Let (a,b,c)=(25, 20, -45),(p,q)=(2, -1/4) the we obtain

u^2 = 99520312500t^4+289912500000t^3+340875000000t^2+198000000000t+54000000000..........(2)

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

Y^2 = X^3-55815X -2917810.

Rank is 1 and generator is (X,Y)=[-6776306/33489,-1852945822/6128487].

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

Numerical example:

[X,Y,Z,W] = [13, 14, 1, 16470].