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The Erdős–Ulam problem on rational distances
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
| 1 | { |
| 2 | "domain": "mathematics", |
| 3 | "topics": [ |
| 4 | "discrete-geometry", |
| 5 | "number-theory" |
| 6 | ], |
| 7 | "type": "proof", |
| 8 | "also": [ |
| 9 | "construction" |
| 10 | ], |
| 11 | "cover": { |
| 12 | "motif": "points", |
| 13 | "n": 8 |
| 14 | }, |
| 15 | "source": { |
| 16 | "key": "erdos", |
| 17 | "name": "Erdős Problems", |
| 18 | "url": "https://www.erdosproblems.com/212", |
| 19 | "id": "212", |
| 20 | "curator": "Thomas Bloom", |
| 21 | "poser": "Stanisław Ulam", |
| 22 | "status": "open", |
| 23 | "as_of": "2026-10-20", |
| 24 | "catalog": "0521c19bbd38" |
| 25 | } |
| 26 | } |