Mid-sized divisors of binomial coefficients

DefinitionErdős Problem 387#✓fully proved

The proposition that there is a real number \(c > 0\) such that for all natural numbers \(n\) and \(k\) with \(1 \le k < n\), the binomial coefficient \(\binom{n}{k}\) has a divisor \(d\) with \(cn < d \le n\).

def conjecture : Prop :=
  ∃ c : ℝ, 0 < c ∧
    ∀ n k : ℕ, 1 ≤ k → k < n →
      ∃ d : ℕ, d ∣ n.choose k ∧ c * (n : ℝ) < (d : ℝ) ∧ d ≤ n

Used by 1

Used byErdős Problem 387 is false

TheoremErdős Problem 387 is false#●ready to prove

There is no real number \(c > 0\) such that for all natural numbers \(n\) and \(k\) with \(1 \le k < n\) the binomial coefficient \(\binom{n}{k}\) has a divisor \(d\) with \(cn < d \le n\).

This follows from Theorem 1.4 of Bui, Naprienko, Pratt and Zaharescu: for arbitrarily large \(k\) there are infinitely many \(n\) such that \(\binom{n}{k}\) has no divisor in the interval \((241\, n \log\log k / \log k,\ n]\).

theorem_wanted conjecture_refuted : ¬ conjecture

Sources

• H. M. Bui, S. Naprienko, K. Pratt, A. Zaharescu, Binomial coefficients with divisors avoiding an interval, arXiv preprint (2026), arXiv:2605.21221, https://arxiv.org/abs/2605.21221.

Uses 1

Statement usesErdős Problem 387

1 fully proved1 ready to prove

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