Problem 123

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claims/: The 5 claim pages of Problem 123, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a,b,c> 1$ 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?

Formulation. Erdős also posed the question under the weaker hypothesis that $a,b,c$ have no common factor. The site's commentary reports the suggestion in the paper that offers the prize ([Er97], [Er97e]; neither is held here), and Burr, Erdős, Graham and Li recall the conjecture of [ErLe96] in that form, for any $k\geq2$ positive integers $a_1,\ldots,a_k$ with $\gcd(a_1,\ldots,a_k)=1$ [BEGL96, Section 3, pp. 137--138], printing every exponent as at least $1$. The answer to that form is no, as the site's commentary shows: with $a=6$, $b=10$, $c=15$, the terms with $k+m>0$ are multiples of $3$ and the remaining terms $10^l$ are $\equiv 1 \pmod 3$, so a representation of an integer $\equiv 2 \pmod 3$ must use two terms $10^l$ and $10^{l'}$, one of which divides the other. The Statement is Erdős and Lewin's conjecture (i) of [ErLe96, p. 840], which asks for three pairwise relatively prime integers; the site's $a,b,c>1$ excludes the base $1$, with which the terms reduce to $b^lc^m$, and the Corollary of [ErLe96, p. 838] shows that those are $d$-complete only for $\{b,c\}=\{2,3\}$. Erdős's stronger conjecture of [Er92b] for $2,3,5$, with all summands in a window $[x,(1+\epsilon)x)$, is discussed with the claims below.

Status. PROVED (LEAN). The site labels the problem PROVED (LEAN); the proof and its acceptance are recorded on the claim pages, and this corpus has built none of the Lean.

Source. erdosproblems.com/123, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #123, https://www.erdosproblems.com/123.

References.

• [BEGL96] Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. 77 (1996), 133-138.
• [ChYu23b] Chen, Yong-Gao and Yu, Wang-Xing, On $d$-complete sequences of integers, II. Acta Arith. (2023), 161-181.
• [Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
• [Er97] Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.
• [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
• [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.
• [MaCh16] Ma, Mi-Mi and Chen, Yong-Gao, On $d$-complete sequences of integers. J. Number Theory (2016), 1-12.

Formalization. Statement in formal-conjectures.

Current assessment

The question. The statement above; PROVED (LEAN), page last edited 17 July 2026. The site's commentary explains that a sequence is $d$-complete when every large integer is a sum of distinct terms no one of which divides another, and that Erdős and Lewin [ErLe96] conjectured this case; the prize is for a proof or disproof. The weaker hypothesis of no common factor, under which the answer is no, is recorded under Formulation.

Claims. Two proofs were posted on the site in July 2026, the first found with GPT 5.6 and the second with GPT 5.6 and Opus 4.8, both with Lean developments that this corpus has not built. The first, Snyder's proof found with GPT 5.6 (2026-07-15), is accepted by the site's curator and settles the problem; its acceptance evidence is that curator review alone. The second, Principia Math's proof (2026-07-20), claims the stronger statement that the summands can be taken from a window $[x,\rho x)$ for any fixed $1<\rho<\min(a,b,c)$, which would also give Erdős's conjecture of [Er92b] for $2,3,5$; the site shows no verdict for it and it stays claimed.

Earlier partial results. Before the proofs, $d$-completeness had been established for particular triples: Erdős and Lewin [ErLe96] proved it for $(3,5,7)$ and for $(2,5,c)$ with $c\in\{7,11,13,17,19\}$; Ma and Chen [MaCh16] extended the base pair $(2,5)$ to further odd $c$ coprime to $10$ under a finite-interval condition; and Chen and Yu [ChYu23b] covered $(2,5,c)$ for $3\leq c\leq 87$, $(2,7,c)$ for $3\leq c\leq 33$, and $(3,5,c)$ for $2\leq c\leq 14$, each $c$ coprime to the other two bases. Each paper is recorded as an accepted partial claim: Erdős and Lewin's triples, Ma and Chen's criterion and triples and Chen and Yu's ranges. The site credits Snyder, not these papers, with settling the problem. The site's commentary lists these; the papers other than [ErLe96] are not held.

Formalization and the Lean label. The formal-conjectures statement file, at its commit of 2026-09-18 (ErdosProblems/123.lean), marks erdos_123 solved and names as its formal proof a copy of Snyder's development in Boris Alexeev's repository, at the commit linked from the claim page; it marks the Erdős--Lewin case $(3,5,7)$ and the two-base variant for $2,3$ solved without proofs, and keeps Erdős's conjecture for $2,3,5$ (erdos_123.variants.powers_2_3_5_snug) open. Neither development has been built, kernel-checked or audited in this corpus, and no statement-fidelity review exists.

Search scope. The site's problem page, commentary and proof-claims page, and the two claimants' repositories for their headers and commit dates. arXiv, Crossref, MathSciNet, zbMATH, Google Scholar and X were not searched.

Remaining gaps. (1) No refereed write-up of either proof exists; the acceptance rests on the curator's review of Snyder's claim. (2) Neither Lean development has been built in this corpus, so neither page lists formalized evidence. (3) The two developments' author headers attribute the first proof to different AI systems (GPT 5.6 on the site's entry, Claude Fable 5 in the repository copy); the discrepancy is recorded on the claim page and not resolved. (4) Erdős's stronger conjecture for $2,3,5$ is claimed only by the pending claim.

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