The board / number theory / #56
Erdős Problem #56
DISPROVED LEANdatabase updated 2025-11-26
The statement as posed is false. 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).
Say a set of natural numbers is `k`-weakly divisible if any `k+1` elements of `A` are not relatively prime.
`MaxWeaklyDivisible N k` is the size of the largest k-weakly divisible subset of `{1,..., N}`
`FirstPrimesMultiples N k` is the set of numbers in `{1,..., N}` that are a multiple of one of the first `k` primes.
Suppose \(A \subseteq \{1,\dots,N\}\) is such that there are no \(k+1\) elements of \(A\) which are relatively prime. An example is the set of all multiples of the first \(k\) primes. Is this the largest such set? To avoid trivial counterexamples, we must insist that \(N\) be at least the \(k\)th prime.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/56.Prize$10
Topicsnumber theory, intersecting family
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/56
Attempts
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