It is a surprising fact that the proportion of integer lattice points visible from the origin is
exactly
,
or approximately 60%. Hence, approximately 40% of the integer lattice is hidden
from the origin. Since 1971, many have studied a variety of problems involving
lattice-point visibility, in particular, searching for patterns in that 40% of the
lattice composed of invisible points. One such pattern is a square patch, an
grid
of
invisible points, which we call a hidden forest. It is known that there exist arbitrarily
large hidden forests in the integer lattice. However, the methods up to now involve
the Chinese remainder theorem (CRT) on the rows and columns of matrices with
prime number entries, and they have only been able to locate hidden forests
very far from the origin. For example, using this method the closest known
hidden forest is over
3 quintillion, or
,
units away from the origin. We introduce the concept of quasiprime matrices and
utilize a variety of computational and theoretical techniques to find some of the
closest known hidden forests to date. Using these new techniques, we find a
hidden forest that is merely 184 million units away from the origin. We conjecture
that every hidden forest can be found via the CRT-algorithm on a quasiprime
matrix.
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