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].

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This page was last edited 28 December 2025.