The board / number theory / #428
Erdős Problem #428
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 density ratio of set \(A\) up to \(n\) relative to the prime counting function \(\pi(n)\).
Is there a set \(A\subseteq \mathbb{N}\) such that, for infinitely many \(n\), all of \(n-a\) are prime for all \(a\in A\) with \(0 < a < n\) and \[\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?\]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/428.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/428
Attempts
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Any hypothetical witness sequence is rigorously forced by a uniform Selberg prime-pair bound to have gaps at least $\exp(m^{c_A})$, an exact synchronized-chain reduction and two-witness maxima through
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