Erdős Problem 684 at Density One: Small-prime Parts of Binomial Coefficients and Gaussian Fluctuations

Eric Li

For $0\le k\le n$, let $u(n,k)$ be the largest divisor of $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ whose prime factors are at most $k$. Erdős Problem #684 concerns the special threshold $u(n,k)>n^2$ and ask…