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Erdős Problem #351

PROVED LEANdatabase updated 2026-05-12

The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.

What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).

The set of rational numbers of the form `P(n) + 1 / n` where `n` is a natural number and `P` is a polynomial with rational coefficients. Note: We include `P 0` in there (since `1 / 0 = 0`), but this doesn't change the validity of the conjecture

The predicate that a set `A` is strongly complete, i.e. that for every finite set `B`, every sufficiently large integer is a sum of elements of the set `A \ B`.

The predicate that the rational polynomial `P` has a complete image.

Let \(p(x) \in \mathbb{Q}[x]\) be a non-constant rational polynomial 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 \(B\), \[\left\{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right\}\] contains all sufficiently large integers?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/351.
Prizenone
Topicsnumber theory, complete sequences
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/351

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