31X^4 + 31Y^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 31X^4 + 31Y^4 + 2Z^4 = W^4 has infinitely many integer solutions.

31X^4 + 31Y^4 + 2Z^4 = W^4......................................................(1)
We use an identity 31(t+1)^4+31(t)^4+2(t^2+t+5)^2 = (9+8t+8t^2)^2...............(2)

So, we look for the integer solutions {Z^2 = t^2+t+5, W^2 = 9+8t+8t^2}..........(3)

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

u^2 = 169k^4-720k^3+1110k^2-720k+169............................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 409x + 2760.
Rank is 1 and generator is [177 , 2340].

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

Numerical example:

[X,Y,Z,W] = [359588405, 1077921484, 1722678185, 2782963573], 
            [33721506026578833818990756, 30705447934671354915397205, 140423388493681615144489025, 170511055330447359425996293]