MathDB

Let

\mathbf{Z}

denote the set of all integers. Find all real numbers

c > 0

such that there exists a labeling of the lattice points

( x, y ) \in \mathbf{Z}^2

with positive integers for which:
[*] only finitely many distinct labels occur, and [*] for each label

i

, the distance between any two points labeled

i

is at least

c^i

.
Proposed by Ricky Liu

Problem Statement