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 find the number of ordered triples of complex numbers (a, b, c) satisfying the given conditions, we need to analyze the equation and constraints provided.

Let's break down the problem step by step:

1. Condition 1: a^3 - b, b^3 - c, and c^3 - a are rational numbers.
This means that the differences between consecutive elements of the ordered triple must be rational. Let's denote these differences as d1, d2, and d3, respectively.
d1 = a^3 - b, d2 = b^3 - c, d3 = c^3 - a

2. Condition 2: 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}]
This is an equation involving the squares and products of the elements of the ordered triple. It represents a relationship between their values.

Now, let's work on simplifying the equation:

Expand the equation:
a^6 + a^2 + b^6 + b^2 + c^6 + c^2 = 2(a^3 * b + b^3 * c + c^3 * a)

Rearrange the terms:
a^6 - 2a^3 * b + b^6 - 2b^3 * c + c^6 - 2c^3 * a + a^2 + b^2 + c^2 = 0

We notice that this equation looks like the sum of cubes:
(a^2 - a * b + b^2) * (a^4 + a^2 * b^2 + b^4 - a - b) + (b^2 - b * c + c^2) * (b^4 + b^2 * c^2 + c^4 - b - c) + (c^2 - c * a + a^2) * (c^4 + c^2 * a^2 + a^4 - c - a) = 0

Since we want to find solutions where a, b, and c are complex numbers, we can conclude that each term in the above sum must equal zero.

Now, we have two conditions upon which we can determine the number of solutions:
Condition 1: The pairwise differences a^3 - b, b^3 - c, and c^3 - a must be rational numbers.
Condition 2: Each of the terms in the sum of cubes factorization must equal zero.

Note that we cannot directly determine the exact number of solutions, but we can proceed as follows:

For Condition 1:
Assume that a, b, and c are rational numbers. In this case, a^3 - b, b^3 - c, and c^3 - a are all rational numbers.
Now, count the number of ordered triples (a, b, c) consisting of rational numbers that satisfy Condition 2.

For Condition 2:
Analyze each of the terms separately and determine the conditions for each factor to be zero.

By further solving the equation and examining the conditions for the factors to be zero, you can proceed to find the possible solutions or determine the number of solutions satisfying both conditions.

Please note that due to the complexity of the problem, it may not be possible to easily determine the number of ordered triples without further calculations and analysis.