Problem 1110

Erdős Problem #1110: declared status 'open'. Formalized: no. Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other. If $\{p,q\}\neq \{2,3\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers? Current best: Yu and Chen [YuCh22] proved$$\frac{n}{(\log n)^{\log_23}}\ll f(n) \ll \frac{n}{\log n}.$$Yang and Zhao [YaZh25] improved the lower bound to $f(n)\gg n/\log n$. Prize: no. Tags: number theory.