Let $2=p_1 k$ many prime factors?

Erdős problem 891 · problems.science

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?

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

891.lean

Retained formal statement2 of 4

This is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?

FormalConjectures/ErdosProblems/891.leanErdos891.erdos_891.variants.case_k_21 lineExact file

1True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ m ∈ Finset.Ico n (n + 6), 3 ≤ ArithmeticFunction.cardDistinctFactors m

OpenStatement only, no proof

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