40x^6 + 24y^6 - 64z^6 = w^2


We show diophantine equation 40x^6 + 24y^6 - 64z^6 = w^2 has infinitely many integer solutions.

40x^6 + 24y^6 - 64z^6 = w^2.....................................................................(1)

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

Let (a,b,c)=(40, 24, -64),(p,q)=(2, -2/3) the we obtain

u^2 = 477395681280t^4+1390663434240t^3+1664719257600t^2+951268147200t+305764761600..............(2)

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

Y^2 = X^3-8103375X -4942653750.

Rank is 1 and generator is (X,Y)=[-2450 , -14300].

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

Numerical example:

[X,Y,Z,W] = [4915, 9907, 6787, 4123125901440].




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