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arXiv · 2605.21221

Binomial coefficients with divisors avoiding an interval

Hung M. Bui·Slava Naprienko·Kyle Pratt·Alexandru Zaharescu
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Abstract

We solve a fifty-year-old conjecture of Erd\H{o}s and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger than a constant times $n$. We show this is the case when $k$ is sufficiently large as a function of $n$. However, we show it is possible to find binomial coefficients ${n \choose k}$, where $k$ is small compared to $n$, such that ${n \choose k}$ does not have divisors $\leq n$ close to $n$. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.

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Hung M. Bui, Slava Naprienko, Kyle Pratt, Alexandru Zaharescu. 2026-05-20. Binomial coefficients with divisors avoiding an interval. https://arxiv.org/abs/2605.21221

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