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Maxwell's Conjecture on Point-Charge Equilibria

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maxwell-conjectureMathematical physicsposed by James Clerk Maxwell, 2004recorded: disproved

1 attempt · no person has looked

Statement

Do nn point charges whose electrostatic potential has only non-degenerate critical points always have at most (n−1)2(n-1)^2 of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges pulled apart into a shallow bipyramid - has at least 24>1624 > 16 non-degenerate critical points, so the conjecture is false.

Context

Traces to Maxwell's 1873 Treatise; the modern form drove work in potential theory and real algebraic geometry.

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1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.6 Sol with Philip Arathoon, Gavin Ball, Matthew D. Kvalheim ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Sol
    people
    Philip Arathoon, Gavin Ball, Matthew D. Kvalheim

    The idea behind the counterexample construction was suggested by the model; the authors verified all mathematical details and wrote the note.

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