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PreprintarXiv licencearXiv

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 Erdős and Graham concerning whether the binomial coefficient binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ 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 binomnk, where k is small compared to n, such that binomnk does not have divisors ≤ 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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Cite

BibTeXRIS
BibTeX
@misc{bui2026binomial,
  title = {{Binomial coefficients with divisors avoiding an interval}},
  author = {Hung M. Bui and Slava Naprienko and Kyle Pratt and Alexandru Zaharescu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.21221},
  eprint = {2605.21221},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.21221},
}
RIS
TY  - UNPB
TI  - Binomial coefficients with divisors avoiding an interval
AU  - Hung M. Bui
AU  - Slava Naprienko
AU  - Kyle Pratt
AU  - Alexandru Zaharescu
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2605.21221
UR  - https://arxiv.org/abs/2605.21221
AB  - We solve a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ 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 binomnk, where k is small compared to n, such that binomnk does not have divisors ≤ n close to n. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.
ER  -