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Omni-MATH / Let 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)∈Z² with positive integers for which: [list] [*] only f…

Problem

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problem

Let Z\mathbf{Z} denote the set of all integers. Find all real numbers c>0c > 0 such that there exists a labeling of the lattice points (x,y)∈Z2 ( x, y ) \in \mathbf{Z}^2 with positive integers for which: [list] [*] only finitely many distinct labels occur, and [*] for each label ii, the distance between any two points labeled ii is at least cic^i. [/list] [i]
Plain-text mathematical notation (without MathML)
Let 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)∈Z² with positive integers for which: 
[list]
[*] only finitely many distinct labels occur, and
[*] for each label i, the distance between any two points labeled i is at least c^(i).
[/list]
[i]
Original LaTeX notation
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: 
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$.
[/list]
[i]

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Source and history

task_id
b93332d9-c4d7-599a-8759-71b184950c64
task_key
test--b93332d9-c4d7-599a-8759-71b184950c64
task_revision_id
1
config / split
/ test

Official source

initial import