47X^4 + 47Y^4 + 2Z^4 = 49W^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 47X^4 + 47Y^4 + 2Z^4 = 49W^4 has infinitely many integer solutions.

47X^4 + 47Y^4 + 2Z^4 = 49W^4..........................................................(1)
We use an identity 47(7(t+1))^4+47(t)^4+2(t^2+49t+49)^2 = 49(48t^2+96t+49)^2..........(2)

So, we look for the integer solutions {Z^2 = t^2+49t+49, W^2 = 48t^2+96t+49}..........(3)

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

u^2 =k^4+190k^2-9024k+106033..........................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 - 27260x + 431808.
Rank is 2 and generator is [-89 , -1470], [352 , -5880].

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

Numerical example:

[X,Y,Z,W] = [952, 475, 2073, 1123], [4592, 735, 6727, 4753]