PROVED (LEAN)
This has been solved in the affirmative and the proof verified in Lean.
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 $k,l\geq 0$.
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 $> n/(\log n)^{1+o(1)}$.
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This page was last edited 07 December 2025.
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