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

31X^4 + 31Y^4 + Z^4 = 2W^4...........................................................(1)
We use an identity 31(8(t+1))^4+31(t)^4+(t^2+64t+64)^2=2(4(63t^2+126t+64))^2.........(2)

So, we look for the integer solutions {Z^2 = t^2+64t+64, W^2 =252t^2+504t+256}.......(3)

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

u^2 =k^4+250k^2-15624k+242173........................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 -3865x + 19175.
Rank is 1 and generator is [95 , -720].

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

Numerical example:

[X,Y,Z,W] = [352, 235, 917, 894],[5379855, 48094816, 16495729, 95443562].