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Erdős Problem #870

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erdos-870Number theoryposed by Paul Erdős, Melvyn Nathanson, 1979recorded: candidate

1 attempt · no person has looked

Statement

Let k≥3k\geq 3 and AA be an additive basis of order kk. Does there exist a constant c=c(k)>0c=c(k)>0 such that if r(n)≥clog⁡nr(n)\geq c\log n for all large nn (where r(n)r(n) counts representations of nn as a sum of at most kk elements of AA) then AA must contain a minimal basis of order kk? The claimed answer is no, for every k≥3k\geq 3.

Context

A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.4 Pro, GPT-5.5 Pro with David Turturean ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.4 Pro, GPT-5.5 Pro
    people
    David Turturean

    The proof was developed via an automated multi-turn scaffold that iteratively queried GPT-5.4 Pro and GPT-5.5 Pro over roughly forty turns, with constructions inspired by the Larsen-Larsen order-2 basis; the author later reworked the k=3 case after community concerns and verified the write-up himself and with GPT-5.5 Pro.

    Reviews

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