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The Lonely Runner Conjecture for Nine and Ten Runners

Number theory · posed by Jörg M. Wills; independently Thomas W. Cusick, 1967 · partial

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Statement

The Lonely Runner Conjecture of Wills and Cusick states that among k+1k+1 runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least 1/(k+1)1/(k+1) 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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. computation · #1

    Tanupat Trakulthongchai, using GPT-5

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    "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.

    Settles 9 and 10 runners only; the general conjecture remains open.

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