19X^4 + 19Y^4 = 2Z^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 19X^4 + 19Y^4 = 2Z^4 + W^4 has infinitely many integer solutions.

19X^4 + 19Y^4 = 2Z^4 + W^4.......................................................(1)

We use an identity 19(t+1)^4+19(t)^4-2(t^2-5t-3)^2 = (1+8t+6t^2)^2...............(2)

So, we look for the integer solutions {Z^2 = t^2-5t-3, W^2 = 1+8t+6t^2}..........(3)

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

u^2 = 961k^4-2736k^3+2910k^2-1368k+239...........................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + 1432 - 5792.
Rank is 2 and generator is [36 , 304], [129 , 1525].

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

Numerical example:

[X,Y,Z,W] = [8209, 46228, 81169, 14397], [774538729, 1172284876, 1797499597, 2161836867].