40x^6 + 8y^6 - 48z^6  = w^2


We show diophantine equation 40x^6 + 8y^6 - 48z^6  = w^2 has infinitely many integer solutions.

40x^6 + 8y^6 - 48z^6  = w^2..........................................................(1)

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

Let (a,b,c)=(40, 8, -48),(p,q)=(2, -4) the we obtain

u^2 = 9248440320t^4-10947133440t^3+10380902400t^2-1258291200t+943718400..............(2)

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

Y^2 = X^3-3243375X -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, 326380320].