36X^4 + Y^4 + 3Z^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 36X^4 + Y^4 + 3Z^4 = W^4 has infinitely many integer solutions.

36X^4 + Y^4 + 3Z^4 = W^4....................................................(1)
We use an identity 36(t+1)^4+(t)^4+3(2t^2+2t)^2 = (6+12t+7t^2)^2............(2)

So, we look for the integer solutions {Z^2 =2t^2+2t, W^2 = 6+12t+7t^2}......(3)

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

u^2 = k^4+24................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 6x.
Rank is 1 and generator is [-2 , -2].

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

Numerical example:

[X,Y,Z,W] = [200, 529, 460, 721], [368082, 3500641, 1605318, 3614885]





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