The board / number theory / #844
Erdős Problem #844
PROVED LEANdatabase updated 2026-04-26
The statement as posed is true, and a proof exists. 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).
Those \(n \in \{1,\ldots,N\}\) which are even or not squarefree, that is, the set of even numbers together with the odd non-squarefree numbers.
Let \(A\subseteq \{1,\ldots,N\}\) be such that, for all \(a,b\in A\), the product \(ab\) is not squarefree. Is the maximum size of such an \(A\) achieved by taking \(A\) to be the set of even numbers and odd non-squarefree numbers? A problem of Erdős and Sárközy. Weisenberg has provided the following positive proof. It is clear that such a maximal \(A\) must contain all non-squarefree numbers. It therefore suffices to find the largest size of a subset of all squarefree numbers in \(\{1,\ldots,N\}\) such that any two have at least one prime factor in common. By the result of Chvátal [Ch74] discussed in [701] this is maximised by the set of all even squarefree numbers. An alternative proof was independently found by Alexeev, Mixon, and Sawin [AMS25]. This was formalized in Lean by Jennings using Aristotle.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/844.Prizenone
Topicsnumber theory, intersecting family
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/844
Attempts
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