The board / number theory / #890
Erdős Problem #890
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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`omegaGt k n` counts the number of distinct prime factors of `n` that are strictly greater than `k`.
If \(\omega_k(n)\) counts the number of distinct prime factors of \(n\) which are \(>k\), then is it true that, for every \(k\geq 1\), \[\liminf_{n\to \infty}\sum_{0\leq i < k}\omega_k(n+i)\leq k?\]
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/890.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/890
Attempts
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Exact identities reduce #890 to a parity-level almost-prime tuple problem and a coefficient-1 maximal-order bound for \(\omega(\prod_{i<k}(n+i))\); an independently rechecked exhaustive scan on \(5{,}
PARTIAL