Erdős Problem #683

Let \(P(n, k)\) be the largest prime factor of \(\binom{n}{k}\). There exists \(c > 0\) such that \(P(n, k) \ge \min(n - k + 1, k^{1 + c})\) for all \(0 < k \le n/2\). Erdős stated this for \(1 \le k \le n\) with the bound \(\min(n-k+1, k^{1+c})\) [Er79d]. The minimum is needed even for \(k \le n/2\): at \(n = 2k\) every prime factor of \(\binom{2k}{k}\) is at most \(2k\), so \(P(n, k) \ge k^{1+c}\) fails for large \(k\). The range \(k \le n/2\) is natural (cf. [#961](https://www.erdosproblems.com/961) and the [discussion](https://www.erdosproblems.com/forum/discuss/683)).