Erdős Problem #684 — For $0\leq k\leq n$ write $\binom{n}{k} = uv$ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.

The problem asks for the largest possible size of the first index $k$ where the small-prime part of a binomial coefficient exceeds $n^2$. The long-standing expectation was $f(n)reaklesssim\log n$; a recent manuscript instead claims arbitrarily large multiples of $\log n$ occur.

Known results

• Mahler proved $f(n)\to\infty$, ineffectively.
• Tang and ChatGPT obtained $f(n)\le n^{30/43+o(1)}$, improved conditionally to $n^{2/3+o(1)}$.
• Alexeev, Putterman, Sawhney, Sellke, and Valiant proved $f(n)\le(24/(\pi^2-6)+o(1))(\log n)^2$.
• The same authors constructed $n_j$ with $f(n_j)\ge(1/2+o(1))\log n_j$.

Unbounded logarithmic limsup: recent claim

The manuscript claims that for every fixed $C>1$, infinitely many $n_M$ satisfy $f(n_M)>(C-o(1))\log n_M$, hence $\limsup_{n\to\infty}f(n)/\log n=\infty$. Its construction uses $n_M=t\operatorname{lcm}(1,\ldots,M)-1$ and a Fourier sieve. This is a claimed resolution at the order level, not yet independently verified.

Current status (as of June 2026): The claimed manuscript would settle the worst-case order by making the logarithmic limsup infinite, while the proof and construction remain unverified; the polylogarithmic upper bound and density-one result are established only as reported results.