27x^6 + 12y^6 - 39z^6 = w^2


We show diophantine equation 27x^6 + 12y^6 - 39z^6 = w^2 has infinitely many integer solutions.

27x^6 + 12y^6 - 39z^6 = w^2..........................................................(1)

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

Let (a,b,c)=(27, 12, -39),(p,q)=(27, 12, -39) the we obtain

u^2 = 5180014476t^4+14124520800t^3+18914575680t^2+9170703360t+3930301440.............(2)

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

Y^2 = X^3-403574535X -1655406435150.

Rank is 2 and generator is (X,Y)=[165370 , -66738430],[-1747750022079/138815524 , 1955766972161970999/1635524503768].

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

Numerical example:

[X,Y,Z,W] = [6761, 4211, 3385, 1608449490648].