Crossed Lines

Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of the number of times that line is crossed by another line in the set. For example, two sets of three lines are shown below:

In both cases $M(L)$ is $3$ and $S(L)$ is $6$: each of the three lines is crossed by two other lines. Note that even if the lines cross at a single point, all of the separate crossings of lines are counted.

Consider points $(T_{2k-1}, T_{2k})$, for integer $k \ge 1$, generated in the following way:

$S_0 = 290797$
$S_{n+1} = S_n^2 \bmod 50515093$
$T_n = (S_n \bmod 2000) – 1000$

For example, the first three points are: $(527, 144)$, $(-488, 732)$, $(-454, -947)$. Given the first $n$ points generated in this manner, let $L_n$ be the set of unique lines that can be formed by joining each point with every other point, the lines being extended indefinitely in both directions. We can then define $M(L_n)$ and $S(L_n)$ as described above.

For example, $M(L_3) = 3$ and $S(L_3) = 6$. Also $M(L_{100}) = 4948$ and $S(L_{100}) = 24477690$.

Find $S(L_{2500})$.

This problem involves a combination of Number Theory, Geometry and Advanced Algorithm Techniques. Given the scope and complexity of the question, it’s currently beyond current state-of-the-art AI techniques to provide a complete solution in real-time.

However, to make an attempt, one would need to start by generating the sequence of points as outlined in the question using the provided formulas for $S_n$ and $T_n$. Once the points are generated, the unique lines can be computed by joining each point with every other point, which could be done using a nested loop. However, as the number of points increase, this approach would become computationally expensive.

Next step would be to calculate $M(L_n)$, which is the number of unique lines. This can be done by storing each unique line in a Set data structure (that automatically filters out duplicates) and then counting the total number of lines in the Set.

Finally, to calculate $S(L_n)$, we would need to calculate the number of times each line in the set is crossed by another line in the set. This is the most challenging part of the question, as it requires determining intersection points of each pair of lines, and may involve non-trivial calculations depending on whether the lines are represented in vector, parametric, or implicit form.

A brute force approach to compute line intersections would also prove too costly for this problem as it would require time proportional to the square of the number of lines for each point. One would need to apply an advanced geometric algorithm (like the Bentley-Ottmann algorithm) to improve the intersection computations.

Unfortunately, the exact solution to this problem can’t be attained directly without programming and use of more elaborate algorithms. This is beyond the capacity of a text-based assistant and would require a programming environment to solve.

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