Let $k\ge 2$ and define $n_k\ge 2k$ to be the least value of $n$ such that $n-i$ divides $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ for all but one $0\le i<k$. Estimate $n_k$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write [ \binom nk=\frac{n(n-1)\cdots(n-k+1)}{k!}. ] For (m=n-i) [[nomath]](with $0\le i<k$)[[/nomath]] we have $ \frac{\binom nk}{m}=\frac{\prod_{0\le j<k,\ j\ne i}(n-j)}{k!}. $ Hence $ m\mid \binom nk \quad\Longleftrightarrow\quad k!\mid \prod_{0\le j<k,\ j\ne i}(n-j). $ Equivalently, in $p$-adic valuations, $ m\mid \…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.