12X^4 + 12Y^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 12X^4 + 12Y^4 = 8Z^4 + W^4 has infinitely many integer solutions.
This equation is related to 3X^4 + 3Y^4 = 2Z^4 + 4W^4.

12X^4 + 12Y^4 = 8Z^4 + W^4......................................................(1)
We use an identity 12(t+1)^4+12(t)^4-8(t^2-t-1)^2 = 4(1+4t+2t^2)^2..............(2)

So, we look for the integer solutions {Z^2 = t^2-t-1, W^2 = 2+8t+4t^2}..........(3)

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

u^2 = -23k^4+12k^3-58k^2-84k+409................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - x^2 + 9x - 9.
Rank is 1 and generator is [3 , 6].

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

Numerical example:

[X,Y,Z,W] = [9, 17, 19, 14], [2937, 1873, 619, 5678].