39x^6 + 13y^6 - 52z^6 = w^2


We show diophantine equation 39x^6 + 13y^6 - 52z^6 = w^2 has infinitely many integer solutions.

39x^6 + 13y^6 - 52z^6 = w^2.............................................................(1)

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

Let (a,b,c)=(39, 13, -52),(p,q)=(2, -2) the we obtain

u^2 = 15812626284t^4+22589466120t^3+56473665300t^2+15059644080t+11294733060..............(2)

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

Y^2 = X^3-3855735X -1474077150.

Rank is 1 and generator is (X,Y)=[-13460722992573450/9480259474009,984723229291976772447780/29189747361252133027].

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

Numerical example:

[X,Y,Z,W] = [3319, 4969, 1247, 497616148848].