73X^4 + 73Y^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 73X^4 + 73Y^4 + Z^4 = 2W^4 has infinitely many integer solutions.

73X^4 + 73Y^4 + Z^4 = 2W^4........................................................(1)
We use an identity 73(t+3)^4+73(t)^4+(4t^2+12t-45)^2 = 2(9t^2+27t+63)^2...........(2)

So, we look for the integer solutions {Z^2 = 4t^2+12t-45, W^2 =9t^2+27t+63}.......(3)

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

u^2 = 301k^4-360k^3+1336k^2-1440k+4816............................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 -1434x -15624.
Rank is 2 and generator is [-471/25, -8892/125],[140 , 1596].

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

Numerical example:

[X,Y,Z,W] = [2361, 5481, 1758, 13587], [887, 383, 298, 2199].