Erdős Problem #379 — Prime-power divisibility in binomial coefficients

Erdős Problem $379$ asks whether the divisibility depth $S(n)$ is unbounded along some sequence of integers $n$. The problem is attributed to Erdős; the source does not state when it was posed.

Affirmative solution (date not stated)

Cambie, Kovač, and Tao proved the assertion, and a short elementary complete proof was subsequently obtained. For $r\ge2$, choosing $n=2^{\phi(p^r)}$ with $p>2^{r-1}$ gives $S(n)\ge r$; hence $\limsup S(n)=\infty$. The argument shows every $\binom{n}{k}$, $1\le k<n$, is divisible by either $2^r$ or $p^r$. The discussion also records simpler constructions involving $3^{2^k}$.

Current status (as of June 2026): The problem is resolved affirmatively; the complete proof establishes $\limsup S(n)=\infty$.