Erdős Problem #1095

declared status 'open'. Formalized: yes. Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$. Current best: 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 Erd\H{o}s, Lacampagne, and Selfridge [ELS93] and Granville and Ramar\'{e} [GrRa96]. The current record is$$g(k) \gg \exp(c(\log k)^2)$$for some $c>0$, due to Konyagin [Ko99b]. Prize: no. OEIS: A003458. Tags: binomial coefficients, number theory.

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