Eisensteins Everywhere

Looking at a random direction from the origin, what is the probability that your line of sight doesn't encounter an Eisenstein integer (that is, the probability that a line from the origin at a random direction does not intersect any Eisenstein integer)?

If the probability is \(P, \) submit your answer as \(\lfloor 1000P \rfloor \).


  • An Eisenstein integer is a complex number of the form \(a+b\omega,\) where \(\displaystyle \omega = \frac{-1+i\sqrt{3}}{2}\) and \(a\) and \(b\) are both integers. For example, \(1, \omega\) and \(3+2\omega\) are all Eisenstein integers.

  • \( \lfloor \cdot \rfloor \) denotes the floor function.



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