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 statement3 of 4

Schinzel deduced from Pólya's theorem [Po18] (that the sequence of $k$ -smooth integers has unbounded gaps) that this is true with $p_1\cdots p_k$ replaced by $p_1\cdots p_{k-1}p_{k+1}$ .

FormalConjectures/ErdosProblems/891.leanErdos891.erdos_891.variants.schinzel4 linesExact file

1∀ k ≥ 2,2  ∀ᶠ (n : ℕ) in Filter.atTop,3    ∃ m ∈ Finset.Ico n (n + (∏ i ∈ Finset.range (k - 1), Nat.nth Nat.Prime i) * Nat.nth Nat.Prime k),4      k < ArithmeticFunction.cardDistinctFactors m

SolvedStatement only, no proof

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