The board / number theory / #534
Erdős Problem #534
SOLVEDdatabase updated 2025-08-31
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A set `A ⊆ {1, …, N}` is *admissible* if it contains `N` and any two distinct elements of `A` have a common factor.
The largest size of an admissible subset of `{1, …, N}`.
The prime factors `q₁ < ⋯ < qⱼ` of `N` which are at most `q`.
What is the largest possible subset \(A\subseteq\{1,\ldots,N\}\) which contains \(N\) such that \(\mathrm{gcd}(a,b)>1\) for all \(a\neq b\in A\)? Erdős conjectured that if \(N=q_1^{k_1}\cdots q_r^{k_r}\) (where \(q_1<\cdots <q_r\) are distinct primes) then the maximum is achieved by, for some \(1\leq j\leq r\), those integers in \([1,N]\) which are a multiple of at least one of \(\{2q_1,\ldots,2q_j,q_1\cdots q_j\}\). This conjecture was proved by Ahlswede and Khachatrian [AhKh96].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/534.Prizenone
Topicsnumber theory, intersecting family
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/534
Attempts
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