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The Lonely Runner Conjecture for Nine and Ten Runners

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lonely-runner-nine-tenNumber theoryposed by Jörg M. Wills; independently Thomas W. Cusick, 1967recorded: 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.

Context

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

The Lonely Runner Conjecture is field-famous across number theory and combinatorics with its own Wikipedia article and six decades of partial results; scored on the underlying conjecture per the methodology.

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1 attempt

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  • #1

    Attempt 1

    computationGPT-5 with Tanupat Trakulthongchai ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    GPT-5
    people
    Tanupat Trakulthongchai

    "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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