Skip to content

Erdős problem 891

Start work

Let 2=p1<p2<⋯2=p_1 < p_2 < \cdots be the primes and k≥2k\geq 2. Is it true that, for all sufficiently large nn, there must exist an integer in [n,n+p1⋯pk)[n,n+p_1\cdots p_k) with >k>k many prime factors?

Sources

The source's own material, retained exactly.

4 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

891.lean

Retained formal statement1 of 4

Let 2=p1<p2<⋯2=p_1 < p_2 < \cdots be the primes and k≥2k\geq 2. Is it true that, for all sufficiently large nn, there must exist an integer in [n,n+p1⋯pk)[n,n+p_1\cdots p_k) with >k>k many prime factors?

FormalConjectures/ErdosProblems/891.leanErdos891.erdos_8914 linesExact file
True ↔  ∀ k ≥ 2,    ∀ᶠ (n : ℕ) in Filter.atTop,      ∃ m ∈ Finset.Ico n (n + ∏ i ∈ Finset.range k, Nat.nth Nat.Prime i), k < ArithmeticFunction.cardDistinctFactors m
OpenStatement only, no proof