23X^4 + 23Y^4 + 2Z^4 = 25W^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 23X^4 + 23Y^4 + 2Z^4 = 25W^4 has infinitely many integer solutions.

23X^4 + 23Y^4 + 2Z^4 = 25W^4...........................................................(1)
We use an identity 23(5(t+1))^4+23(t)^4+2(t^2+25t+25)^2 = 25(24t^2+48t+25)^2...........(2)

So, we look for the integer solutions {Z^2 = t^2+25t+25, W^2 = 24t^2+48t+25}...........(3)

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

u^2 =k^4+94k^2-2208k+12697.............................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + x^2 - 3358x + 26288.
Rank is 1 and generator is [-22 , -300].

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

Numerical example:

[X,Y,Z,W] = [360, 43, 457, 371], [27466465, 305184600, 227991965, 300897175]




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