X^4 + 9Y^4 + 6Z^4 = W^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 X^4 + 9Y^4 + 6Z^4 = W^4 has infinitely many integer solutions.

X^4 + 9Y^4 + 6Z^4 = W^4......................................................(1)
We use an identity (2(t+1))^4+9(2t)^4+6(4t^2+4t)^2 = 16(1+2t+4t^2)^2.........(2)

So, we look for the integer solutions {Z^2 = 4t^2+4t, W^2 =4+8t+16t^2}.......(3)

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

u^2 = 3k^4+16................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 12x.
Rank is 1 and generator is [-2 , -4].

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

Numerical example:

[X,Y,Z,W] = [1, 4, 2, 7], [121, 9, 33, 122]




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