Problem #1095: Let $g(k)>k+1$ be the smallest $n$ such that all prime...

---

Progress

A question of Ecklund, Erdős, and Selfridge [EES74], who proved\[k^{1+c}<g(k)\leq \exp((1+o(1))k)\]for some constant $c>0$, and conjectured $g(k)<L_k=[1,\ldots,k]$, the least common multiple of all integers $\leq k$, for all large $k$. In [EES74] they further conjecture that\[\limsup \frac{g(k+1)}{g(k)}=\infty\]and\[\liminf \frac{g(k+1)}{g(k)}=0.\]The lower bound was improved by Erdős, Lacampagne, and Selfridge [ELS93] and Granville and Ramaré [GrRa96]. The current record is\[g(k) \gg \exp(c(\log k)^2)\]for some $c>0$, due to Konyagin [Ko99b].

Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.

Sorenson, Sorenson, and Webster [SSW20] give heuristic evidence that\[\log g(k) \asymp \frac{k}{\log k}.\]See also [1094].