57X^4 + 57Y^4 = 8Z^4 + W^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 57X^4 + 57Y^4 = 8Z^4 + W^4 has infinitely many integer solutions.

57X^4 + 57Y^4 = 8Z^4 + W^4.........................................................(1)
We use an identity 57(t+4)^4+57(2t)^4-8(4(t^2+9t+4))^2 = (-112-24t+29t^2)^2........(2)

So, we look for the integer solutions {Z^2 = 4t^2+36t+16, W^2 =112+24t-29t^2}......(3)

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

u^2 = 7k^4-12k^3-118k^2+1092k-2453.................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - x^2 + 3184x + 29466.
Rank is 2 and generator is [71 , 780],[43 , 494].

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

Numerical example:

[X,Y,Z,W] = [4, 3, 7, 1],[3720292, 786657, 1725881, 10210709].