Problem Statement
Let $(a,b)=1$. The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$.
Categories:
Number Theory
Progress
Proved by Birch [Bi59]. This also follows from a later more general result of Cassels [Ca60] (see [254]).Davenport observed in [Bi59] that this is still true even with $l\ll_{a,b}1$. Hegyvári [He00b] gave an explicit upper bound for this threshold (which is quadruple exponential in $a$ and $b$). This was improved to triply exponential by Fang and Chen [FaCh17].
Yu [Yu24] showed that one can write any large $n$ as the sum of distinct integers of this form, all of which are $> n/(\log n)^{1+o(1)}$.
Source: erdosproblems.com/246 | Last verified: January 14, 2026