Erdős Problem #387
Prior state unknown→disproved
false in general; the positive direction holds for k large relative to n
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number-theory / Number theory
Erdős Problem #387
Erdős and Graham asked whether $\binom{n}{k}$ with $1 \le k \le n/2$ must always have a divisor $\le n$ that is close to $n$, meaning bigger than a fixed constant times $n$. Settled in both directions: true when $k$ is large enough as a function of $n$, but false in general, since there are $\binom{n}{k}$ with $k$ small compared to $n$ having no such divisor.
35Significance / 100
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Erdős and Graham asked whether $\binom{n}{k}$ with $1 \le k \le n/2$ must always have a divisor $\le n$ that is close to $n$, meaning bigger than a fixed constant times $n$. Settled in both directions: true when $k$ is large enough as a function of $n$, but false in general, since there are $\binom{n}{k}$ with $k$ small compared to $n$ having no such divisor.
false in general; the positive direction holds for k large relative to n
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