The board / number theory / #254
Erdős Problem #254
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
An integer `n` can be written as a sum of distinct elements of `A`.
Let \(A\subseteq \mathbb{N}\) be such that \(\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty\) and \(\sum_{n\in A} \{ \theta n\}=\infty\) for every \(\theta\in (0,1)\), where \(\{x\}\) is the distance of \(x\) from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of \(A\).
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/254.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/254
Attempts
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Live page status OPEN; it lists 2 claimed proofs and Steve_Fan as currently working, so the mandatory no-duplication/no-collision rule applies.
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