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Thread: Lattice Points

  1. #1
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    Lattice Points

    Hey, the following is meant to be an application of the Chinese Remainder Theorem.

    A lattice point in the plane is a point $\displaystyle (m,n)$, where m and n are integers.
    A point $\displaystyle (m,n) \in \mathbb{Z}^2$ is called $\displaystyle invisible$ if the straight line segment from $\displaystyle (0,0)$ to $\displaystyle (m,n)$ goes through some other integer lattice point.
    Show that for every $\displaystyle M > 0$ there exists a square in $\displaystyle \mathbb{Z}^2$ of side length M, so that all its lattice points are invisible.
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  2. #2
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    Quote Originally Posted by EinStone View Post
    Hey, the following is meant to be an application of the Chinese Remainder Theorem.

    A lattice point in the plane is a point $\displaystyle (m,n)$, where m and n are integers.
    A point $\displaystyle (m,n) \in \mathbb{Z}^2$ is called $\displaystyle invisible$ if the straight line segment from $\displaystyle (0,0)$ to $\displaystyle (m,n)$ goes through some other integer lattice point.
    Show that for every $\displaystyle M > 0$ there exists a square in $\displaystyle \mathbb{Z}^2$ of side length M, so that all its lattice points are invisible.
    EDIT:
    Found the proof here, so the problem is solved.
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