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

12X^4 + 3Y^4 + Z^4 = W^4..........................................................(1)

We use an identity 12(t+1)^4+3(t+2)^4+(t^2+4t+2)^2 = 4(2t^2+5t+4)^2...............(2)

So, we look for the integer solutions {Z^2 = t^2+4t+2, W^2 = 4t^2+10t+8}..........(3)

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

u^2 = 43k^4-168k^3+326k^2-336k+151................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 -x^2 - 19x - 17.
Rank is 2 and generator is [-2 ,-3], [7 ,12].

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

Numerical example:

[X,Y,Z,W] = [17, 11, 7, 32], [17753, 29963, 24487, 44636].