The board / number theory / #332
Erdős Problem #332
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).
The set of numbers \(D(A)\) which occur infinitely often as \(a_1 - a_2\) with \(a_1, a_2 \in A\).
A set \(S \subseteq \mathbb{Z}\) has bounded gaps if it is syndetic, meaning there is a uniform bound \(M\) such that every interval of length \(M\) contains an element of \(S\).
Let \(A\subseteq \mathbb{N}\) and \(D(A)\) be the set of those numbers which occur infinitely often as \(a_1 - a_2\) with \(a_1, a_2\in A\). What conditions on \(A\) are sufficient to ensure \(D(A)\) has bounded gaps? The sufficient conditions are the unknown `erdos_332.answer`. In order to solve this problem one has to provide what the sufficient conditions are, and proof that they imply the desired condition. If the condition is a solution to the problem is up to human judgement.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/332.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/332
Attempts
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Proved an explicit zero-upper-Banach-density realization \(D(A)=S\) for every \(S\subseteq\mathbb N\), independently verified positive-upper-Banach sufficiency and the packing reduction, and ran the s
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