5X^4 + 45Y^4 + Z^4 = 6W^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 5X^4 + 45Y^4 + Z^4 = 6W^4 has infinitely many integer solutions.

5X^4 + 45Y^4 + Z^4 = 6W^4.....................................................(1)
We use an identity 5((t+1))^4+45(2t)^4+(t^2-10t+1)^2=6(11t^2+1)^2.............(2)

So, we look for the integer solutions {Z^2 = t^2-10t+1, W^2 =11t^2+1}.........(3)

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

u^2 =k^4+42k^2+440k+1101......................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 -312x + 1184.
Rank is 1 and generator is [28 , 120].

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

Numerical example:

[X,Y,Z,W] = [23, 56, 47, 93],[204217, 244776, 722063, 520997].