The Lonely Runner Conjecture for Nine and Ten Runners
The Lonely Runner Conjecture of Wills and Cusick states that among runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least from all others. Following Rosenfeld's computer-assisted proof for 8 runners, the paper refines his approach with a sieve and proves the cases of 9 and 10 runners.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-assisted
- Method
- Computation
- Field
- Diophantine approximation
- Posed by
- Jörg M. Wills; independently Thomas W. Cusick
- Year posed
- 1967
- Years open
- 58y
- Solved
- 2025-11-27
- Model
- GPT-5
- Vendor
- OpenAI
- Collaborators
- Tanupat Trakulthongchai
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Settles 9 and 10 runners only; the general conjecture remains open.
What the AI did
"We employed OpenAI GPT-5 to assist with code generation, especially in low-level optimization" of the C++ verification that constitutes the proof; code and result receipts are public.
Verification
A computer-assisted proof in the Rosenfeld tradition; the code and receipts are public but no independent rerun or review has appeared.