17X^4 + 17Y^4 = 18Z^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 17X^4 + 17Y^4 = 18Z^4 + W^4 has infinitely many integer solutions.

17X^4 + 17Y^4 = 18Z^4 + W^4..........................................................(1)

We use an identity 17(t+2)^4+17(t+5)^4-18(t^2+3t-4)^2 = (4t^2+46t+103)^2.............(2)

So, we look for the integer solutions {Z^2 = t^2+3t-4, W^2 = 4t^2+46t+103}...........(3)

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

u^2 = 153k^4-36k^2-17................................................................(4)

This quartic equation is birationally equivalent to an elliptic curve below.
y^2 = x^3 + 9972x - 253152.
Rank is 1 and generator is [124 , 1700].

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

Numerical example:

[X,Y,Z,W] = [409, 457, 399, 934], [85904137, 76960153, 87076399, 148575958].






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