Problem 1110

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claims/: The 3 claim pages of Problem 1110, one per claimant's result; the problem's standing derives from them.

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Statement. Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.

If $\{p,q\}\neq \{2,3\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?

Formulation. The site asks whether there are infinitely many "coprime non-representable numbers", and Erdős and Lewin (p. 840) also say only coprime. The phrase reads either as infinitely many non-representable numbers coprime to $pq$ or as an infinite pairwise coprime family of non-representable numbers. The pairwise reading implies the other, since each prime factor of $pq$ divides at most one member of a pairwise coprime family. The formal-conjectures statement takes the pairwise reading. The page's standing targets the site's wording, and each claim page that bears on the second question says which reading it covers.

Status. Open. The site's proof-claims tab carries three partial proof claims, each recorded on its own claim page and none adopted here: a claim of 2026-08-05 by the forum user Apiros3 that each of the three pairs Yu and Chen left open has an infinite pairwise-coprime sequence of non-representable numbers prime to $pq$ (claim page); a claim of 2026-08-24 by Ding, Li, Liu and Zhang that the representable integers for $(5,2)$ have positive lower density (claim page); and a claim of 2026-09-29 by Bécart of the same for $(7,2)$, with a repository draft of the same date for $(4,3)$ (claim page). The site labels the problem open (page last edited 1 April 2026; proof-claims thread accessed 2026-10-06).

Source. erdosproblems.com/1110, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1110, https://www.erdosproblems.com/1110.

References.

• [BMS98] Blecksmith, Richard and McCallum, Michael and Selfridge, J. L., $3$-smooth representations of integers. Amer. Math. Monthly (1998), 529-543.
• [Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
• [ErLe96] Erdős, P. and Lewin, Mordechai, [[../library/diophantine_problems/erdos_1996_d_complete_sequences_integers/_index|$d$-complete sequences of integers]]. Math. Comp. (1996), 837-840.
• [YaZh25] Yang, Quan-Hui and Zhao, Lilu, A conjecture of Yu and Chen related to the Erd\H os-Lewin theorem. Acta Arith. (2025), 277-286.
• [YuCh22] Yu, Wang-Xing and Chen, Yong-Gao, On a conjecture of Erdős and Lewin. J. Number Theory 238 (2022), 763-778.

Formalization. Statement in formal-conjectures, which reads the second question as an infinite pairwise coprime family of non-representable integers, in the category research open with no formal proof named.

Current assessment

The site's formulation poses two questions for coprime $p>q\geq 2$ with $\{p,q\}\neq\{2,3\}$: what can be said about the density of the non-representable numbers, and whether there are infinitely many coprime non-representable numbers. Both questions are the ones Erdős and Lewin put on p. 840 of Erdős and Lewin 1996, after their Theorem 1 and its Corollary showed that every integer is representable only for $\{p,q\}=\{2,3\}$ and that the non-representable numbers are closed under multiplication by $p$ and $q$. The site's remarks record that Erdős [Er92b] had conjectured the $\{2,3\}$ case and that it has a short induction proof. In the literature the site cites, Yu and Chen [YuCh22] proved that the representable numbers have density zero when $q>3$, when $q=3$ and $p>6$, and when $q=2$ and $p>10$, and that there are infinitely many coprime non-representable numbers when $q>3$, when $q=3$ and $p\neq 5$, and when $q=2$ and $p\notin\{3,5,9\}$. The site also records a threshold question of Erdős and Lewin for the pair $\{2,3\}$: with $f(n)$ the fastest-growing function such that every large $n$ is a sum of numbers $2^k3^l$, none dividing another, all exceeding $f(n)$, Yu and Chen proved $n/(\log n)^{\log_2 3}\ll f(n)\ll n/\log n$, Yang and Zhao [YaZh25] raised the lower bound to $n/\log n$, and a comment on the site observes that a result of Blecksmith, McCallum and Selfridge [BMS98] already gives $f(n)\sim\tfrac{1}{2}(\log 2)(\log 3)\,n/\log n$. Problem 123, Problem 845 and Problem 246 treat three bases, the $\{2,3\}$ density question and the problem without the non-divisibility condition.

Three pending partial claims, all unreviewed, bear on the two questions. Apiros3 2026 claims that each of the three pairs $(5,2)$, $(9,2)$ and $(5,3)$ has an infinite pairwise-coprime sequence of non-representable numbers prime to $pq$, the second question in its stronger reading, and its Lean development also proves Yu and Chen's range, so the development asserts the second question for every coprime pair other than $\{2,3\}$. Ding, Li, Liu and Zhang 2026 and Bécart 2026 claim that the representable integers have positive lower density for $(5,2)$ and $(7,2)$, two pairs outside Yu and Chen's density-zero range, and Bécart's repository adds a draft of the same for $(4,3)$, a third such pair, so that the first question's answer differs between pairs. None of the three would settle the problem alone, none is accepted, and the derived standing is open. The search behind this account, made 2026-10-07, covered the site's page, its proof-claims thread, the claimants' postings and the formal-conjectures statement file; the statements attributed above to Yu and Chen [YuCh22] and to Yang and Zhao [YaZh25] follow the site's remarks, not the papers.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1996_d_complete_sequences_integers
• erdos_1992_my_favourite_problems_various_branches_combinatorics