Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 351

claims/: The 3 claim pages of Problem 351, one per claimant's result; the problem's standing derives from them.

Statement. Let $p(x)\in \mathbb{Q}[x]$ with positive leading coefficient. Is it true that

A=\{ p(n)+1/n : n\in \mathbb{N}\}

is strongly complete, in the sense that, for any finite set $B$ ,

\left\{\sum_{n\in X}n : X\subseteq A\backslash B\textrm{ finite }\right\}

contains all sufficiently large integers?

Status. Proved, in the site's label "PROVED (LEAN)". The argument that GPT 5.5 Pro produced for Problem 283, posted on 2026-05-03 by Liam Price and edited by Kevin Barreto, yields the statement for every $p$ with positive leading coefficient; Nat Sothanaphan confirmed it with ChatGPT, it is formalized in Lean, and the site accepted it (page last edited 10 May 2026). See the claim page. Earlier partial results, each with its own claim page: Graham [Gr63] for $p(x)=x$ (accepted, refereed) and van Doorn's note of 2025-09-15 deducing $p(x)=x^2$ from Graham's method and Alekseyev [Al19] (claimed). The (Lean) suffix of the site's label PROVED (LEAN) means nothing here: this corpus has not built or audited the Lean proof.

Source. erdosproblems.com/351, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #351, https://www.erdosproblems.com/351.

References.

[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose reciprocals sum to 1. (2019), 213-221.
[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.
[Gr64f] Graham, R. L., Complete sequences of polynomial values. Duke Math. J. (1964), 275-285.

Formalization. Statement in formal-conjectures at the file's last change (2026-09-18), whose formal_proof attribute points to a Lean wrapper of the Problem 283 development; the claim page pins the proof files.

Progress

Not yet compiled.

Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (14)

Graham: the case p(x) = x is strongly complete Van Doorn: the case p(x) = x^2 is strongly complete Strong completeness from the Problem 283 argument problems/additive_bases/E0351/claims Additive Bases and Sidon Sets Problem 283 Polynomial sums over Egyptian fractions for every positive rational Additive Bases and Sidon Sets additive_bases/graham_1964_complete_sequences_polynomial_values unit_fractions/alekseyev_2019_partitions_into_squares_distinct_integers_whose Theorem 1: 8542 is the largest integer not a sum of squares of distinct integers with reciprocal sum 1 unit_fractions/graham_1963_theorem_partitions Theorem 1: every integer above 77 is a sum of distinct integers whose reciprocals sum to 1 Theorem 2: partitions into distinct integers above m with reciprocal sum 1

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