Binomial coefficients with divisors avoiding an interval
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917516582846464 |
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| author | Bui, Hung M. Pratt, Kyle Zaharescu, Alexandru |
| author_facet | Bui, Hung M. Pratt, Kyle Zaharescu, Alexandru |
| contents | We investigate a fifty-year-old conjecture of Erdős 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 (under the Generalized Riemann Hypothesis) 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 settles the conjecture of Erdős and Graham, under GRH. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Binomial coefficients with divisors avoiding an interval Bui, Hung M. Pratt, Kyle Zaharescu, Alexandru Number Theory We investigate a fifty-year-old conjecture of Erdős 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 (under the Generalized Riemann Hypothesis) 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 settles the conjecture of Erdős and Graham, under GRH. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates. |
| title | Binomial coefficients with divisors avoiding an interval |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.21221 |