25X^4 + 25Y^4 + 8Z^4 = 25W^4

Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number.
So, we are curious about whether above equation has a solution or not if abcd is not square number.
In particular, when does this equation have infinitely many integer solutions?

Parametric solutions of x^4 + ay^4 = z^4 + bt^4 are given below.
x^4 + ay^4 = z^4 + bt^4

We show diophantine equation 25X^4 + 25Y^4 + 8Z^4 = 25W^4 has infinitely many integer solutions.

25X^4 + 25Y^4 + 8Z^4 = 25W^4................................................(1)
We use an identity 25(t+1)^4+25(2t)^4+8(5t^2+5t)^2 = 25(1+2t+5t^2)^2........(2)

So, we look for the integer solutions {Z^2 = 5t^2+5t, W^2 =1+2t+5t^2}.......(3)

By parameterizing the second equation and substituting the result to first equation, then we obtain quartic equation below.

u^2 = 4k^4+25...............................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 25x.
Rank is 1 and generator is [-4 , -6].

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

Numerical example:

[X,Y,Z,W] = [9, 40, 30, 41], [2307361, 2420640, 3736740, 3344161]