Problem 123

Erdős Problem #123: declared status 'open'. Formalized: yes. Let $a,b,c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\geq 0$), none of which divide any other? Current best: As a partial record of progress so far, the sequence $\{a^kb^lc^m\}$ is known to be $d$-complete when: {UL} {LI}$a=3$, $b=5$, $c=7$ (Erd\H{o}s and Lewin [ErLe96]).{/LI} {LI}$a=2$, $b=5$, $c\in \{7,11,13,17,19\}$ (Erd\H{o}s and Lewin [ErLe96]).{/LI} {LI}$a=2$, $b=5$, $c\in \{9,21,23,27,29,31\}$ - more generally, $a=2$, $b=5$, and any $c>6$ with $(c,10)=1$ such that there exists $N$ where every integer in $(N,25cN)$ is the sum of distinct elements of $\{2^k3^lc^m\}$, none of which divide any other (Ma and Chen [MaCh16]).{/LI} {LI} $a=2$, $b=5$, $3\leq c\leq 87$ with $(c,10)=1$, or $a=2$, $b=7$, $3\leq c\leq 33$ with $(c,14)=1$, or $a=3$, $b=5$, $2\leq c\leq 14$ with $(c,15)=1$ (Chen and Yu [ChYu23b]).{/LI} {/UL} In [Er92b] Erd\H{o}s makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\cdots <b_t$ of the form $2^k3^l5^m$ where $b_t<(1+\epsilon)b_1$. Prize: $250. Tags: number theory.