8. Given that there are 2017 pairwise disjoint circles on the blackboard, and no three circles have a common tangent. A tangent segment of two circles is defined as the line segment connecting the two points of tangency of one of their common tangents. Luciano draws tangent segments on the blackboard one by one, ensuring that each tangent segment does not intersect any other circle and does not intersect any previously drawn tangent segment. He continues to draw tangent segments according to this rule until he can no longer draw any more. Find all possible values of the number of tangent segments Luciano has drawn when he stops.

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
8. If there are $n$ circles, then the number of tangent segments is exactly
$3(n-1)$. Therefore, the number of tangent segments drawn on the blackboard is $3(2017-1)=6048$.
First, consider the case where the centers of the circles $C_{1}, C_{2}, \cdots, C_{n}$ are collinear, and each circle $C_{i}$ is partially obscured by the adjacent circles. For example:

Let the center of circle $C_{i} (i=1,2, \cdots, n)$ be $\left(i^{2}, 0\right)$, and the radius be $\frac{i}{2}$. Then, tangent segments can only be drawn between adjacent circles $C_{i}$ and $C_{i+1} (i=1,2, \cdots, n-1)$, and exactly three tangent segments can be drawn between any two adjacent circles.

Thus, in this case, Luciano can draw exactly $3(n-1)$ tangent segments, as shown in Figure 13.
For the general case, the approach from the initial state to the final state (where no more tangent segments can be drawn) involves continuously adjusting the sizes and positions of the circles so that no four circles have a common tangent, and proving that the number of drawn tangent segments remains constant as the configuration changes. Using this method, any configuration of circles and tangent segments can be adjusted to the special case mentioned earlier, thus the number of tangent segments in the final state is always $3(n-1)$.

First, consider all possible tangent segments for any two circles. A tangent segment that is tangent to a circle $A$ can be defined in two possible directions: leaving circle $A$ in a clockwise direction or in a counterclockwise direction. Two tangent segments that are tangent to the same circle and have the same direction never intersect. For any pair of circles $(A, B)$, there are four tangent segments that can be determined by their directions. For example, the tangent segment $(A+, B-)$ indicates a segment that leaves circle $A$ in a clockwise direction and leaves circle $B$ in a counterclockwise direction. If we do not consider intersections, there are a total of $4 \mathrm{C}_{n}^{2}=2 n(n-1)$ possible tangent segments.

Choose a circle $C$ and start continuously moving and adjusting the size of circle $C$ while keeping the directions of all existing tangent segments related to circle $C$ unchanged. The adjustments can be made until a "transition" occurs in the configuration: assume that the distance between circle $C$ and other circles is at least $\varepsilon$, where $\varepsilon$ is a fixed positive constant, with no loss of generality. Thus, under this transition, there are two possible scenarios:
(1) A tangent segment $t$ is suddenly blocked;
(2) An un-drawn tangent segment $t$ suddenly becomes unblocked and can be drawn.
The following lemma is proved:
Lemma: A transition occurs only when three circles $C_{1}, C_{2}, C_{3}$ have a common tangent line $l$, i.e., the tangent segments between these three circles (all on $l$) are not blocked by any other circle or tangent segment different from $C_{1}, C_{2}, C_{3}$.

Proof: In fact, since (2) is the inverse of (1), it suffices to prove (1).

Assume that the segment $t$ is suddenly blocked. Consider the following two cases.
(i) $t$ is blocked by a circle.
As shown in Figure 14.
A new circle becomes the third circle tangent to $l$, and no other circle or tangent segment blocks $t$.
(ii) $t$ is blocked by another tangent segment $t'$.
When two tangent segments $t$ and $t'$ intersect for the first time, it must be at the endpoint of one of the segments. If one endpoint of $t'$ first crosses the interior of $t$, the circle associated with this endpoint has already blocked $t$ (i.e., case (i)), so it can only be the case where $t$ and $t'$ have a common endpoint. If this happens, the common endpoint must belong to a single circle (considering that the distance between different circles is at least $\varepsilon$, except during mutual translation and size adjustment), thus, for this circle, they must have different directions.

Therefore, at the moment of transition, $t$ and $t'$ are tangent to the same point on the same circle, and they must lie on the same line $l$, as shown in Figure 15.

Thus, three circles are simultaneously tangent to $l$, and no other circle or tangent segment blocks $t$ or $t'$ (otherwise, they would not exist before the transition).
The lemma is proved.
Next, consider the maximum number of tangent segments immediately before and after a transition (i.e., when three circles have a common tangent line $l$).

Let the three circles $C_{1}, C_{2}, C_{3}$ be ordered by their points of tangency. The tangent segments that may be affected are $t_{12}, t_{23}, t_{13}$ (for $1 \leqslant i<j \leqslant 3$, $t_{ij}$ represents the tangent segment between circles $C_{i}$ and $C_{j}$). Since circle $C_{2}$ is in the middle, $t_{12}$ and $t_{23}$ must have different directions for circle $C_{2}$. For circle $C_{1}$, $t_{12}$ and $t_{13}$ must have the same direction, and for circle $C_{3}$, $t_{13}$ and $t_{23}$ must have the same direction. Figure 16 illustrates this situation, where one can choose either circle $C_{1}$ or $C_{1}'$, and either circle $C_{3}$ or $C_{3}'$, to slightly adjust the configuration so that the three circles no longer have a common tangent line, while maintaining the directions of $t_{12}, t_{23}, t_{13}$. No other circle or tangent segment can block any of these three tangent segments, and tangent segments with the same direction and endpoint on the same circle never block each other. Thus, a simple configuration can be used to verify that the tangent segments can be drawn.
(i) $t_{13}$ passes through circle $C_{2}$.
In this case, $t_{13}$ cannot be drawn, while $t_{12}$ and $t_{23}$ can be drawn, as shown in Figure 17.
(ii) $t_{13}$ does not pass through circle $C_{2}$.
In this case, $t_{13}$ can be drawn, while $t_{12}$ and $t_{23}$ block each other, so only one tangent segment can be drawn, as shown in Figure 18.
Since in each case exactly two out of the three tangent segments can be drawn, the maximum number of tangent segments remains constant when the circles are translated and resized.
Therefore, the conclusion to be proved holds.