Problem Statement
Estimate $n_k$, the smallest integer $>2k$ such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.
Categories:
Number Theory
Progress
Erdős and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.In [Er79d] Erdős writes that probably $n_k<e^{o(k)}$ but $n_k>k^d$ for all constant $d$.
Adenwalla observes that an easy upper bound is $n_k\leq \prod_{k<p<2k}p=e^{O(k)}$.
Source: erdosproblems.com/451 | Last verified: January 15, 2026