# 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 $$\lfloor 1000P \rfloor$$.

Notes:

• 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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