Runners for which the lonely runner conjecture is proved
the largest such that the lonely runner conjecture holds for every runners
- steps
- 7
- by AI
- 0
- since
- 1972
The line is the frontier over time, climbing as the bound goes up: higher is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the bottom edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
Wills in 1967, and Cusick independently in 1974, conjectured that if runners start together on a circular track of unit length at pairwise distinct constant speeds, then each runner is at some time at distance at least from all the others. Trivial for , it has fallen one or a few runners at a time: 4 in 1972, 5 in 1984, 6 in 2001, 7 in 2008, then, after a seventeen-year pause, a computer-assisted method built on Tao's finite-checking reduction settled 8 in September 2025 and has been extended almost monthly since, to 9 and 10, and to 13 by April 2026. The conjecture for all is open; this frontier tracks how far the case-by-case verification has reached.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 26 Apr 2026 | best Eleven, twelve and thirteen runners, extending the same computational method with parallel execution and stronger pruning. The paper says its code development was AI-assisted and prints a proof sketch it attributes to ChatGPT-5.6 Pro, so this is an AI-in-the-loop result without a catalog entry yet. Drawn as a cited row until it has one. | Touch Sungkawichai and Tanupat Trakulthongchai, with AI-assisted code and one lemma sketched by ChatGPT-5.6 Pro | – | historical | source ↗ |
| 1 Dec 2025 | Nine runners, independently, four days after the entry's nine and ten, by improvements to his own eight-runner method. On the reviewed track this is the record until April 2026. | Matthieu Rosenfeld | – | historical | source ↗ |
| 27 Nov 2025 | Nine and ten runners, by adding a sieve to Rosenfeld's verification; GPT-5 assisted the C++ implementation, per the paper's Section 6. Code and result receipts are public; no independent rerun has appeared, so the entry is unreviewed and this row is a candidate. | Tanupat Trakulthongchai, with GPT-5 | GPT-5 | candidate unreviewed | entry |
| 17 Sept 2025 | The method every later row uses: Tao's 2018 reduction to a finite check, sharpened by Malikiosis, Santos and Schymura in 2025, then a computer verification. Rosenfeld's abstract predicted that minor improvements would reach 9 or 10; they did within ten weeks. | Matthieu Rosenfeld | – | historical | source ↗ |
| 2008 | The lonely runner with seven runners, Electronic Journal of Combinatorics 15 (2008). The record for seventeen years. | Javier Barajas and Oriol Serra | – | historical | source ↗ |
| 2001 | Six lonely runners, Electronic Journal of Combinatorics 8 (2001). Seventeen years after five. | Tom Bohman, Ron Holzman and Daniel Kleitman | – | historical | source ↗ |
| 1984 | View-obstruction problems III, Journal of Number Theory 19 (1984). Computer-assisted at the time; Bienia, Goddyn, Gvozdjak, Sebo and Tarsi gave an elementary proof in 1998. | Thomas W. Cusick and Carl Pomerance | – | historical | source ↗ |
| 1972 | Monatshefte fur Mathematik 76 (1972), in the Diophantine-approximation language Wills posed the conjecture in five years earlier. The cases n <= 3 are elementary. | Ulrich Betke and Jorg M. Wills | – | historical | source ↗ |
Rows from the Wikipedia article's per-n account and its works-cited list, each reference then checked at Crossref or arXiv; the four 2025-2026 papers were opened and their abstracts and AI disclosures read. Rosenfeld's 9 (1 December 2025) came four days after the entry's 9 and 10 and is drawn as a row that did not move the line, since the entry's row is a candidate: unreviewed, so hollow. The April 2026 paper reaching 13 discloses AI-assisted code and a lemma sketched by ChatGPT-5.6 Pro; it is drawn as a cited historical row because it has no catalog entry yet, and it should get one.