25X^4 + 4Y^4 + Z^4 = 5W^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 + 4Y^4 + Z^4 = 5W^4 has infinitely many integer solutions.

25X^4 + 4Y^4 + Z^4 = 5W^4........................................................(1)
We use an identity 25(2(t+1))^4+4(t)^4+(t^2+10t+10)^2=5(9t^2+18t+10)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+10t+10, W^2 =9t^2+18t+10}.......(3)

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

u^2 = k^4+34k^2-288k+577.........................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 -168x + 468.
Rank is 1 and generator is [2 , -12].

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

Numerical example:

[X,Y,Z,W] = [2, 3, 7, 5],[3570, 157, 5393, 5597].