12x^6 + 3y^6 - 15z^6  = w^2


We show diophantine equation 12x^6 + 3y^6 - 15z^6  = w^2 has infinitely many integer solutions.

12x^6 + 3y^6 - 15z^6  = w^2.................................................(1)

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

Let (a,b,c)=(12, 3, -15),(p,q)=(2, -3) the we obtain

u^2 = 2143260t^4-1574640t^3+4592700t^2+656100...............................(2)

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

Y^2 = X^3-1488375X -339558750.

Rank is 1 and generator is (X,Y)=[-1050,8100].

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

Numerical example:

[X,Y,Z,W] = [251, 971, 395, 1568579040].




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