Short proofs in combinatorics and number theory

We give a triplet of short proofs, each of which answers a question raised by Erd\H{o}s. The first concerns the small prime factors of $\left(\genfrac{}{}{0px}{}{n}{k}\right)$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i+A_i$ has bounded gaps, and the final concerns whether $\{\alpha p\}$ is “well-distributed” in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI.