75X^4 + 3Y^4 + 2Z^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 75X^4 + 3Y^4 + 2Z^4 = 5W^4 has infinitely many integer solutions.

75X^4 + 3Y^4 + 2Z^4 = 5W^4...................................................(1)
We use an identity 75(t+1)^4+3(t)^4+2(t^2+5t+5)^2 = 5(4t^2+8t+5)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+5t+5, W^2 =4t^2+8t+5}.......(3)

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

u^2 = k^4+14k^2-48k+37.......................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - x^2 - 13x + 22.
Rank is 1 and generator is [1 , -3].

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

Numerical example:

[X,Y,Z,W] = [3, 11, 1, 10], [5, 19, 31, 26].