20x^6 + 5y^6 - 25z^6  = w^2


We show diophantine equation 20x^6 + 5y^6 - 25z^6  = w^2 has infinitely many integer solutions.

20x^6 + 5y^6 - 25z^6  = w^2.................................................(1)

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

Let (a,b,c)=(20, 5, -25),(p,q)=(3, -7) the we obtain

u^2 = 9418750000t^4-7425000000t^3+3187500000t^2-375000000t+93750000..........(2)

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

Y^2 = X^3-6615X -100610.

Rank is 1 and generator is (X,Y)=[-3106/49,-86778/343].

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

Numerical example:

[X,Y,Z,W] = [4, 6, 2, 560].




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