The board / number theory / #891
Erdős Problem #891
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(2=p_1 < p_2 < \cdots\) be the primes and \(k\geq 2\). Is it true that, for all sufficiently large \(n\), there must exist an integer in \([n,n+p_1\cdots p_k)\) with \(>k\) many prime factors?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/891.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/891
Attempts
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Exact smooth-centre reduction; all k=2 starts through 10^30 and k=3 starts through 10^12 classified, with last bad starts 4372 and 8615 respectively, but no uniform all-large-n lemma.
PARTIAL