Problem 351

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?

Source: erdosproblems.com/351

Number theory
Complete sequences

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

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 (Price Barreto, 2026). 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.

Problem 351

Status

Proof claims

Discussion