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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FormalConjectures/ErdosProblems/

891.lean

Retained formal statement4 of 4

Weisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\cdots p_k$ with $p_1\cdots p_k-1$ . Indeed, let $L_k$ be the lowest common multiple of all integers at most $p_1\cdots p_k$ . By Dickson's conjecture [Wikipedia], there are infinitely many $n'$ such that $\frac{L_k}{m}n'+1$ is prime for all $1\leq m < p_1\cdots p_k$ . It follows that, if $n=L_kn'+1$ , then all integers in $[n,n+p_1\cdots p_k-1)$ have at most $k$ prime factors.

FormalConjectures/ErdosProblems/891.leanErdos891.erdos_891.variants.weisenberg3 linesExact file

1∀ k ≥ 2,2  ∃ᶠ (n : ℕ) in Filter.atTop,3    ∀ m ∈ Finset.Ico n (n + ∏ i ∈ Finset.range k, Nat.nth Nat.Prime i - 1), ArithmeticFunction.cardDistinctFactors m ≤ k

OpenStatement only, no proof

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