How many ordered triples of complex numbers(a,b,c) are there such that a^3-b,b^3-c,c^3-a are rational numbers, and

a^2(a^4+1)+b^2(b^4+1)+c^2(c^4+1)=2[{(a^3) b}+{(b^3)c}+{(c^3)a}]

To solve this problem, we will work through it step by step.

Step 1: Let's rewrite the given conditions in a more manageable form.
We have three conditions:
1. a^3 - b is a rational number.
2. b^3 - c is a rational number.
3. c^3 - a is a rational number.

Step 2: Rewrite the given equation to a more useful form.
The given equation can be rewritten as:
a^6 + a^2 + b^6 + b^2 + c^6 + c^2 = 2(a^3b + b^3c + c^3a)

Step 3: Introduce a new variable.
Let's introduce a new variable, x, which will represent the sum of a^2, b^2, and c^2. Therefore, we have:
x = a^2 + b^2 + c^2

Step 4: Rewrite the equation using the new variable.
Using the new variable, the given equation can be rewritten as:
(x^3 - 3x(a^2 + b^2 + c^2) + 2(a^3b + b^3c + c^3a)) - (a^2 + b^2 + c^2) = 0

Step 5: Simplify the expression inside the parentheses.
The expression inside the parentheses can be simplified as follows:
(x^3 - 3x(a^2 + b^2 + c^2) + 2(a^3b + b^3c + c^3a)) - (a^2 + b^2 + c^2)
= x^3 - 3x(a^2 + b^2 + c^2) + 2(a^3b + b^3c + c^3a) - (a^2 + b^2 + c^2)
= x^3 - 3xa^2 - 3xb^2 - 3xc^2 + 2a^3b + 2b^3c + 2c^3a - a^2 - b^2 - c^2

Step 6: Simplify further using the given conditions.
Using the given conditions that a^3 - b, b^3 - c, and c^3 - a are rational numbers, we can replace them in the expression above:
x^3 - 3xa^2 - 3xb^2 - 3xc^2 + 2a^3b + 2b^3c + 2c^3a - a^2 - b^2 - c^2
= x^3 - 3xa^2 - 3xb^2 - 3xc^2 + 2(a(a^3 - b) + b(b^3 - c) + c(c^3 - a)) - (a^2 + b^2 + c^2)
= x^3 - 3xa^2 - 3xb^2 - 3xc^2 + 2a^4 - 2ab + 2b^4 - 2bc + 2c^4 - 2ac - a^2 - b^2 - c^2

Step 7: Simplify further using the new variable.
Using the new variable x = a^2 + b^2 + c^2, we can rewrite the expression again:
x^3 - 3xa^2 - 3xb^2 - 3xc^2 + 2a^4 - 2ab + 2b^4 - 2bc + 2c^4 - 2ac - a^2 - b^2 - c^2
= x^3 - 4xa^2 - 4xb^2 - 4xc^2 + 2(a^4 + b^4 + c^4) - 2(ab + bc + ac) - (a^2 + b^2 + c^2)
= x^3 - 4x(a^2 + b^2 + c^2) + 2(a^4 + b^4 + c^4) - 2(ab + bc + ac) - (a^2 + b^2 + c^2)
= x^3 - 4x^2 + 2(a^4 + b^4 + c^4) - 2(ab + bc + ac) - x

Step 8: Simplify further using the x variable.
Using the x variable, the expression can be rewritten as:
x^3 - 4x^2 + 2x^2 - 2(ab + bc + ac) - x
Simplifying this, we get:
x^3 - 2x^2 - 2(ab + bc + ac) - x

Step 9: Solve the equation.
Now that we have the equation in a simplified form, we can proceed to solve it. However, it seems challenging to find an exact solution for the equation. We need additional information or assumptions to find a specific solution.