79X^4 + 79Y^4 + 4Z^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 79X^4 + 79Y^4 + 4Z^4 = 2W^4 has infinitely many integer solutions.

79X^4 + 79Y^4 + 4Z^4 = 2W^4........................................................(1)
We use an identity 79(t+2)^4+79(t)^4+4(t^2+2t+22)^2=2(9t^2+18t+40)^2...............(2)

So, we look for the integer solutions {Z^2 = t^2+2t+22, W^2 =9t^2+18t+40}..........(3)

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

u^2 = 67k^4-360k^3+910k^2-360k+67..................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 -10274x + 355656.
Rank is 1 and generator is [121, 948].

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

Numerical example:

[X,Y,Z,W] = [13, 3, 37, 47],[747837, 1026077, 1092307, 2771321].