שמור ב:
| Main Authors: | , , |
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| פורמט: | Preprint |
| יצא לאור: |
2026
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| נושאים: | |
| גישה מקוונת: | https://arxiv.org/abs/2605.21221 |
| תגים: |
הוספת תג
אין תגיות, היה/י הראשונ/ה לתייג את הרשומה!
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תוכן הענינים:
- 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.