Erdős Problem #351

The problem

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?

Statement quoted from erdosproblems.com/351, maintained by Thomas Bloom. Original source: [ErGr80,p.58]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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