The board / number theory / #434
Erdős Problem #434
PROVED LEANdatabase updated 2026-02-24
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).
A natural \(n\) is representable as a set \(A\) if it can be written as the sum of finitely many elements of \(A\) (with repetition allowed).
The number of naturals that cannot be written as the sum of finitely many elements of the set \(A\), with repetition allowed.
Let \(k \le n\). What choice of \(A\subseteq\{1, \dots, n\}\) (with \(\text{gcd}(A) = 1\)) of size \(|A| = k\) maximises the number of integers not representable as the sum of finitely many elements from \(A\) (with repetitions allowed)? Is it \(\{n, n - 1, \dots, n - k + 1\}\)? The maximal choice is indeed \(\{n, \dots, n - k + 1\}\), as proved by Kiss [Ki02]. The Lean theorem assumes \(2 \le k\). When \(k = 1 < n\), the proposed singleton \(\{n\}\) does not have gcd \(1\); the gcd condition instead forces the unique admissible choice \(A = \{1\}\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/434.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/434
Attempts
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