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 statement3 of 4

Schinzel deduced from Pólya's theorem [Po18] (that the sequence of kk-smooth integers has unbounded gaps) that this is true with p1⋯pkp_1\cdots p_k replaced by p1⋯pk−1pk+1p_1\cdots p_{k-1}p_{k+1}.

FormalConjectures/ErdosProblems/891.leanErdos891.erdos_891.variants.schinzel4 linesExact file
∀ k ≥ 2,  ∀ᶠ (n : ℕ) in Filter.atTop,    ∃ m ∈ Finset.Ico n (n + (∏ i ∈ Finset.range (k - 1), Nat.nth Nat.Prime i) * Nat.nth Nat.Prime k),      k < ArithmeticFunction.cardDistinctFactors m
SolvedStatement only, no proof