50X^4 + 2Y^4 + 5Z^4 = 2W^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 50X^4 + 2Y^4 + 5Z^4 = 2W^4 has infinitely many integer solutions.

50X^4 + 2Y^4 + 5Z^4 = 2W^4....................................................(1)
We use an identity 50(t+1)^4+2(t)^4+5(2t^2+2t)^2 = 2(5+10t+6t^2)^2............(2)

So, we look for the integer solutions {Z^2 = 2t^2+2t, W^2 = 5+10t+6t^2}.......(3)

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

u^2 =k^4+20...................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 - 5x.
Rank is 1 and generator is [-1 , -2].

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

Numerical example:

[X,Y,Z,W] = [72, 1, 12, 161], [20449, 64082, 51194, 78723]





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