OPEN
This is open, and cannot be resolved with a finite computation.
Let $p(x)\in \mathbb{Q}[x]$. 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?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
Graham
[Gr63] proved this is true when $p(n)=n$. Erdős and Graham also ask which rational functions $r(x)\in\mathbb{Z}(x)$ force $\{ r(n) : n\in\mathbb{N}\}$ to be complete?
Graham
[Gr64f] gave a complete characterisation of which polynomials $r\in \mathbb{R}[x]$ are such that $\{ r(n) : n\in \mathbb{N}\}$ is complete.
In the comments van Doorn has noted that a positive solution for $p(n)=n^2$ follows from
[Gr63] together with result of Alekseyev
[Al19] mentioned in
[283].
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This page was last edited 02 December 2025. View history
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Formalised statement?
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Additional thanks to: Vjekoslav Kovac and Wouter van Doorn
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #351, https://www.erdosproblems.com/351, accessed 2026-04-11
