Problem 1110

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?

Source: erdosproblems.com/1110

Number theory

Wiki page

Status

Open

No claim settles this problem.

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 (Apiros3, 2026)); 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 (Ding, Li, Liu and Zhang, 2026)); 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 (Bécart, 2026)). The site labels the problem open (page last edited 1 April 2026; proof-claims thread accessed 2026-10-06).

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, $d$ -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.

Problem 1110

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Proof claims

Discussion