Problem Statement
Let $C>0$. Is it true that the set of integers of the form\[n=b_1+\cdots+b_t\textrm{ with }b_1<\cdots<b_t\]where $b_i=2^{k_i}3^{l_i}$ for $1\leq i\leq t$ and $b_t\leq Cb_1$ has density $0$?
Categories:
Number Theory
Progress
In [Er92b] Erdős wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.'Indeed, one proves (by induction) the stronger fact that such a representation always exists, and moreover if $n$ is even then all the summands can be taken to be even (this is actually not stronger, see the comment by marinov): if $n=2m$ we are done applying the inductive hypothesis to $m$. Otherwise if $n$ is odd then let $3^k$ be the largest power of $3$ which is $\leq n$ and apply the inductive hypothesis to $n-3^k$ (which is even).
van Doorn and Everts [vDEv25] have disproved this with $C=6$ - in fact, they prove that all integers can be written as such a sum in which $b_t<6b_1$.
If $C$ is the smallest constant for which all large integers can be written as such a sum with $b_t<Cb_1$ then Erdős and Lewin [ErLe96] show that $C>2$, and van Doorn and Everts [vDEv25] show that $C\geq 3$. In the comments Alexeev and Cambie give evidence that perhaps any $C>3^{10}/2^{14}\approx 3.604$ suffices.
See also [123] and [1110].
Source: erdosproblems.com/845 | Last verified: January 16, 2026