3X^4 + 75Y^4 + 5Z^4 = 8W^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 3X^4 + 75Y^4 + 5Z^4 = 8W^4 has infinitely many integer solutions.

3X^4 + 75Y^4 + 5Z^4 = 8W^4....................................................(1)
We use an identity 3(2(t+2))^4+75(t+1)^4+5(t^2+2t+5)^2 = 8(4t^2+11t+11)^2.....(2)

So, we look for the integer solutions {Z^2 = t^2+2t+5, W^2 = 4t^2+11t+11}.....(3)

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

u^2 =70k^4-300k^3+440k^2-180k+34..............................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 -x^2 -78x+ 252.
Rank is 2 and generator is [3 , -6],[12 , 30].

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

Numerical example:

[X,Y,Z,W] = [22, 15, 17, 28],[96338, 1335, 99017, 98038]





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