Bui 2026 binomial coefficients divisors avoiding interval

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corollary_1_6: The paper's corollary of Theorem 1.4: infinitely many binom(n,k) with k of order (log log n)^{1/2} have no divisor in (n·log log log log n/log log log n, n]; the edition read prints no constant before the ratio.

question_1_1: The Erdős-Graham question as the paper poses it: whether one positive constant c gives every binomial coefficient binom(n,k) with 1 <= k < n a divisor in the interval (cn, n].

theorem_1_2: Bui, Naprienko, Pratt and Zaharescu's theorem that for small fixed ε > 0 and n large in terms of ε, every binom(n,k) with exp((log n)^{2/3+ε}) <= k <= n/2 has a divisor in (n - n/(log n)^{1/4}, n].

theorem_1_4: Bui, Naprienko, Pratt and Zaharescu's theorem that for every large fixed k_0 and small δ > 0 infinitely many binom(n,k) with k_0 < k <= δ(log log n)^{1/2} have no divisor in (n·241 log log k/log k, n].

theorem_5_1: The paper's covering theorem: for large K and 2 <= B <= log K/(240 log log K) some k ~ K and a residue class mod N_k make binom(n,k) free of primes <= k and a product of factors (n-i)/g_i with every g_i >= B.

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Hung M. Bui, Slava Naprienko, Kyle Pratt, Alexandru Zaharescu, Binomial coefficients with divisors avoiding an interval. arXiv:2605.21221 (2026). The copy read for this card is arXiv version v2 (30 Jun 2026); page numbers below are its pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2605.21221), every other right reserved.

The paper settles a fifty-year-old question of Erdős and Graham (Question 1.1, p. 2): is there a positive constant c such that every binomial coefficient binom(n,k) with 1 <= k < n has a divisor in the interval (cn, n]? Theorem 1.2 (pp. 2--3) shows the answer is yes when k is large relative to n: for small ε > 0 and n large, if exp((log n)^{2/3+ε}) <= k <= n/2 then binom(n,k) has a divisor in (n - n/(log n)^{1/4}, n]. Theorem 1.4 (p. 3) gives the negative answer in general: for every sufficiently large fixed constant k_0 and every sufficiently small constant δ > 0 there are infinitely many binom(n,k) with k_0 < k <= δ(log log n)^{1/2} having no divisor in (n·241 log log k / log k, n], and Corollary 1.6 records infinitely many such binom(n,k) with k of order (log log n)^{1/2} and the window (n·log log log log n / log log log n, n]. As this version prints it the corollary has no constant before the ratio; Theorem 1.4 gives the window for such k only with a constant near 482 there. The proof of Theorem 1.4 splits into a covering problem for residue classes (Theorem 5.1) and a divisor problem, the latter handled by sieve methods and exponential sum bounds, including short incomplete Kloosterman sums and Weyl differencing. Together the theorems locate a threshold k_0(n) with (log log n)^{1/2} << k_0(n) << exp((log n)^{2/3+o(1)}) (p. 4), and the paper reports (p. 3) that Erdős later expected a negative answer to Question 1.1, which Theorem 1.4 confirms.

Source: https://arxiv.org/abs/2605.21221.

Bears on. #387: Question 1.1 (p. 2) is the problem's question. Theorem 1.4 (p. 3) gives, for every $c>0$ (taking $k_0$ large in terms of $c$), infinitely many $\binom nk$ with no divisor in $(cn,n]$, which answers it in the negative. Theorem 1.2 (pp. 2--3) gives the opposite in the range $\exp((\log n)^{2/3+\epsilon})\le k\le n/2$, for small $\epsilon>0$ and $n$ large in terms of $\epsilon$: a divisor in $(n-n/(\log n)^{1/4},n]$.

Results.

• Question 1.1 (p. 2): is there $c>0$ such that every $\binom nk$ with $1\le k<n$ has a divisor in $(cn,n]$?
• Theorem 1.2 (pp. 2--3), with Remark 1.3: for small $\epsilon>0$ and $n$ large in terms of $\epsilon$, if $\exp((\log n)^{2/3+\epsilon})\le k\le n/2$ then $\binom nk$ has a divisor in $(n-n/(\log n)^{1/4},n]$.
• Theorem 1.4 (p. 3), with Remark 1.5: for every sufficiently large fixed $k_0$ and sufficiently small $\delta>0$, infinitely many $\binom nk$ with $k_0<k\le\delta(\log\log n)^{1/2}$ have no divisor in $(n\cdot241\log\log k/\log k,\,n]$.
• Corollary 1.6 (p. 3): infinitely many $\binom nk$ with $k\asymp(\log\log n)^{1/2}$ have no divisor in $(n\log\log\log\log n/\log\log\log n,\,n]$, as printed without a constant factor.
• Theorem 5.1 (p. 14), with Definition 5.3 and Proposition 5.5 (p. 15): the covering theorem, a progression of $n$ on which $\binom nk$ has no prime factor $\le k$ and splits as $\prod(n-i)/g_i$ with every $g_i\ge B$.

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