Problem #123: Let $a,b,c$ be three integers which are pairwise coprime

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Progress

A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other. This particular case of $d$-completeness was conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.

As a partial record of progress so far, the sequence $\{a^kb^lc^m\}$ is known to be $d$-complete when:

• $a=3$, $b=5$, $c=7$ (Erdős and Lewin [ErLe96]).


• $a=2$, $b=5$, $c\in \{7,11,13,17,19\}$ (Erdős and Lewin [ErLe96]).


• $a=2$, $b=5$, $c\in \{9,21,23,27,29,31\}$ - 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]).


• $a=2$, $b=5$, $3\leq c\leq 87$ 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]).

In [Er92b] Erdős 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 $b_t<(1+\epsilon)b_1$.

See also [845], and [1110] for the case of two powers.