Erdős Problem #213
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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The predicate (on \(n\)) that there exist \(n\) points in \(\mathbb{R}^2\), no three on a line and no four on a circle, such that all pairwise distances are integers.
Let \(n \geq 4\). Are there \(n\) points in \(\mathbb{R}^2\), no three on a line and no four on a circle, such that all pairwise distances are integers?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/213.Attempts
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The published diameter-22,270 integral heptagon has no distinct eighth point whose seven distances to it are positive integers at most 96,000. Three exact exhaustive traversals and a big-integer metric audit certify the bounded nonextension. This concerns one fixed heptagon and gives no lower bound for arbitrary eight-point sets.
PARTIAL