Erdős problem 387 · problems.science

Is there an absolute constant $c > 0$ such that, for all $1 \leq k < n$ , the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn, n]$ ?

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4 retained statements2415f78e850a

FormalConjectures/ErdosProblems/

387.lean

Retained formal statement3 of 4

Is it true for any $c < 1$ and all $n$ sufficiently large, for all $1 \leq k < n$ , $\binom{n}{k}$ has a divisor in $(cn, n]$ ?

This variant appears in [Gu04]. Bui, Naprienko, Pratt, and Zaharescu [BNPZ26] answered it negatively.

FormalConjectures/ErdosProblems/387.leanErdos387.erdos_387.variants.guy1 lineExact file

1False ↔ ∀ c < 1, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (k : ℕ), 1 ≤ k → k < n → ∃ d, ↑d ∈ Set.Ioc (c * ↑n) ↑n ∧ d ∣ n.choose k

SolvedStatement only, no proof