Problem 123 — Visual explainer (undergrad) ========================================== Problem statement --------------- Let $a,b,c\geq 1$ 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? Picture ------- Target integer N as "prime + offset": N = p + a ^ ^ prime pick a from A Think: A is a small "menu of offsets" that helps you hit all large N. Question: what to look at ------------------------ - Restate the problem as: given the constraint, what asymptotic bound or structure must follow? - Use the comments as benchmarks (best known bounds / constructions). Context ------- Area hint: additive number theory (primes) / sieve methods. Benchmarks / known results (from comments) ---------------------------------------- - 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 Erd\H{o}s and Lewin, 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: {UL} {LI}$a=3$, $b=5$, $c=7$ (Erd\H{o}s and Lewin ).{/LI} {LI}$a=2$, $b=5$, $c\in \{7,11,13,17,19\}$ (Erd\H{o}s and Lewin ).{/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 ).{/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 ).{/LI} {/UL} In 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