The board / number theory / #646
Erdős Problem #646
PROVED LEANdatabase updated 2026-03-02
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).
Let \(p_1,\ldots,p_k\) be distinct primes. Are there infinitely many \(n\) such that \(n!\) is divisible by an even power of each of the \(p_i\)? The answer is yes, proved by Berend [Be97], who further proved that the sequence of such \(n\) has bounded gaps (where the bound depends on the initial set of primes).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/646.Prizenone
Topicsnumber theory, factorials
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/646
Attempts
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