24x^6 + 16y^6 - 40z^6 = w^2


We show diophantine equation 24x^6 + 16y^6 - 40z^6 = w^2 has infinitely many integer solutions.

24x^6 + 16y^6 - 40z^6 = w^2.............................................................(1)

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

Let (a,b,c)=(24, 16, -40),(p,q)=(2, -1/2) the we obtain

u^2 = 25102909440t^4+73232547840t^3+86822092800t^2+50331648000t+15099494400.............(2)

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

Y^2 = X^3-1218375X -292241250.

Rank is 1 and generator is (X,Y)=[-950,2800].

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

Numerical example:

[X,Y,Z,W] = [8777, 15497, 11465, 11905660005120].