Problem 678

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claims/: The 1 claim page of Problem 678, one per claimant's result; the problem's standing derives from them.

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Statement. Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$.

Are there infinitely many $m,n$ and $k\geq 3$ with $m\geq n+k$ such that

$$M(n,k)>M(m,k+1)?$$

Formulation. The site's wording of 2026-09-18 (page last edited 11 January 2026): infinitely many triples $(m,n,k)$ with $k\ge3$ and $m\ge n+k$, so $k$ varies. Erdős's 1979 sentence (Math. Mag., item 2, printed p. 67), "Suppose that $k\ge3$ and $m\ge n+k$. Observe that then, for each $k$, $M(n;k)>M(m;k)$ has infinitely many solutions. Yet I cannot decide whether the same is true for $M(n;k)>M(m;k+1)$", can also be read with $k$ fixed; that reading is false (for each $k$ only finitely many pairs $(m,n)$ occur, since $M(m,k+1)\ge m+1$ bounds $m$ by $M(n,k)$ and the inequality reverses for large $n$; the thread's argument of 6 January 2026, formalized as the variants below), so the site takes the varying-$k$ reading, which, it says, is how the question appears, less ambiguously, in Erdős's 1992 Eureka note [Er92e, p. 47], not held. This page takes the site's wording as the target; the fixed-$k$ reading is a Formulation note on the 1979 item, not a defect of the site's question. The site's source keys are [Er79, p. 67] and [Er92e, p. 47], with [Ca24] cited in the commentary.

Status. Proved; the site's label is PROVED (LEAN). Cambie's Theorem 1 (arXiv:2410.09138v1, 11 October 2024, 5 pages): for every constant $C\ge1$ and every sufficiently large $k$ there are integers $0<x<y$ with $y>x+k$ and $\operatorname{lcm}\{x,\ldots,x+k-1\}>C\cdot\operatorname{lcm}\{y,\ldots,y+k\}$. With $n=x-1$ and $m=y-1$ (an authored one-line substitution), this reads $M(n,k)>C\cdot M(m,k+1)$ with $m>n+k$, so for $C=1$ every large $k$ gives a triple and there are infinitely many; the ratio can moreover exceed any constant. The status-defining source is an arXiv preprint with no journal acceptance found on 2026-09-18; the site accepted it (PROVED (LEAN), 11 January 2026), and an external Lean formalization of the theorem, which imports an external project for the prime number theorem, accompanies it. The suffix is a catalog label explained under Formalization and the Lean label, and no local kernel credit is claimed. Claim page: Cambie 2024 (accepted on the site's documented acceptance; not refereed).

Source. erdosproblems.com/678, accessed 2026-09-18: the problem page (PROVED (LEAN), with the site's note that the answer is yes and the proof verified in Lean; last edited 11 January 2026; header keys [Er79, p.67], [Er92e, p.47]; commentary citing [Ca24] and Problem 677), its three-comment discussion thread (6 to 8 January 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #678, https://www.erdosproblems.com/678, accessed 2026-09-18.

References.

• [Ca24] Cambie, S., Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138v1 (11 October 2024; the text dated October 15, 2024; no journal reference), 5 pages; Theorem 1, p. 2; proof, pp. 2--4. Library home: cambie_2024_resolution_erdos_problem_least_common_multiples; result page Theorem 1.
• [Er79] Erdős, P., Some unconventional problems in number theory. Math. Mag. 52 (1979), no. 2, 67--70; item 2, printed pp. 67--68. Library home: erdos_1979_unconventional_problems_number_theory_math_mag.
• [Er92e] Erdős, P., Some unsolved problems in geometry, number theory and combinatorics. Eureka 52 (1992), 44--48; the site cites p. 47. Not held: the publishing society's web archive index and the magazine's landing page returned HTTP 404 on 2026-09-18.
• [OEIS] The site lists "Possible", no sequence.

Formalization. Statement only here. The file ErdosProblems/678.lean of formal-conjectures,(the commit the link pins), declares erdos_678 : answer(True) ↔ ∀ᶠ k in atTop, {(m, n) | n + k ≤ m ∧ lcmInterval m (k + 1) < lcmInterval n k}.Nonempty and the variants erdos_678.variants.infinitely_many_triples ({(k, m, n) | 3 ≤ k ∧ n + k ≤ m ∧ lcmInterval m (k + 1) < lcmInterval n k}.Infinite) and erdos_678.variants.not_infinitely_many_pairs (¬ ∀ᶠ k in atTop, {(m, n) | …}.Infinite, "why it cannot be asked of a single $k$"), all three under category research solved with proof sorry and a formal_proof attribute naming src/latest/ErdosProblems/Erdos678.lean of plby/lean-proofs at its commit of 1 August 2026, the version the claim page links; four category test lemmas check the referee's examples $M(96,7)>M(104,8)$, $M(132,7)>M(139,8)$ and Cambie's $M(52,7)>M(62,8)$, $M(36,8)>M(47,9)$ by decide (the last lemma's docstring prints $M(48,9)$ where its statement, correctly, has $47$; recomputed here). The community database, records the problem as proved since 7 January 2026, formal_status Lean, the statement formalized since 31 August 2025, no formal-proof URL and OEIS "possible". The site's "(Lean)" suffix is a catalog label; the external file is described under "Formalization and the Lean label" below and was not built here.

Current assessment

The question (site formulation of 2026-09-18). The statement above, labeled PROVED (LEAN) with the site's standard sentence for a positive answer whose proof is verified in Lean, last edited 11 January 2026. The commentary, in this page's words: the referee of [Er79], whom the site says [Er92e] names as Selfridge, found the two solutions $M(96,7)>M(104,8)$ and $M(132,7)>M(139,8)$; the 1979 wording can also be read with $k$ fixed, asking for infinitely many pairs $(m,n)$ for each $k\ge3$, but that reading is easily refuted (the thread's argument, below), and Erdős presumably meant the varying-$k$ question, the form in which, the site says, the 1992 note states it less ambiguously; the answer is yes, by Cambie [Ca24], in the strong form that every sufficiently large $k$ admits $m\ge n+k$ with $M(n,k)>M(m,k+1)$. The commentary then lists the adjacent questions: $M(n,k)>M(m,k)$ has infinitely many solutions, and for $n_k$, the least $n$ with this property, Erdős could prove $n_k/k\to\infty$ but had no good upper bound; and whether $M(t,k)\le M(T,k)$ whenever $t<\min(u_k,T)$, where $u_k$ is the least integer with $M(u_k,k)>M(u_k+1,k)$, to which the commentary reports many counterexamples found by Cambie and van Doorn, for example $k=7$, $u_7=7$, $M(6,7)=M(7,7)>M(8,7)$ (recomputed here: $360360=360360>180180$, and $u_7=7$). The thread, oldest first: a comment of 6 January 2026 (the account BorisAlexeev) that the statement then on the page, which fixed $k\ge3$ before asking for infinitely many $m,n$, matches Erdős's wording, that Aristotle, Harmonic's automated prover, had disproved that fixed-$k$ statement unaided for $k=3$ (no solutions at all), that the argument ($\operatorname{lcm}(n+1,n+2,n+3)\le(n+1)(n+2)(n+3)$ against $\operatorname{lcm}(m+1,\ldots,m+4)\ge\binom{m+4}4$) works for every $k$, and that Cambie's result gives infinitely many triples, after which the site's statement was updated; a comment of the same day (the account Nat Sothanaphan) that ChatGPT confirmed the fixed-$k$ disproof generalizes and that the commenter had checked the short argument by hand; and a comment of 8 January 2026 (the account BorisAlexeev) announcing Aristotle's formalization of Cambie's main result, the commenter's first posted proof that was conditional, assuming a prime number theorem statement pi_alt taken from an external Lean project, with four end results (the main theorem, infinitely many triples, the site's statement, and the negation of the fixed-$k$ statement then in the formal-conjectures collection, the last proved without the prime number theorem), plus a formalization of the whole paper that still had two type-checking errors. The proof-claim tab is empty.

The origin (Er79, printed pp. 67--68). Item 2, after the equality conjectures of Problem 677, poses the question in three sentences: "Suppose that $k\ge3$ and $m\ge n+k$. Observe that then, for each $k$, $M(n;k)>M(m;k)$ has infinitely many solutions. Yet I cannot decide whether the same is true for $M(n;k)>M(m;k+1)$" (item 2, p. 67), followed by a parenthesis giving the referee's two solutions, $M(96;7)>M(104;8)$ and $M(132;7)>M(139;8)$. The rest of the item, paraphrased here: writing $n_k$ for the least solution of $M(n;k)>M(m;k)$, Erdős asks for an estimate of $n_k$ and expects $n_k/k\to\infty$, which a note added in proof confirms as easy while recording that he has no good upper bound for $n_k$; writing $u_k$ for the least integer with $M(u_k;k)>M(u_k+1;k)$, he states as easy that $u_k=(1+o(1))k$ and $u_k>k$, and he could not prove his guess that $t<u_k$ and $T>t$ force $M(t;k)\le M(T;k)$ (p. 68). The two referee examples hold ($8321670749700>3803928503760$ and $71657610814440>35640326882160$, recomputed here). [Er92e] is not held; the site's reading that the 1992 note states the varying-$k$ question and names the referee is second-hand here. Status-defining source. Theorem 1 of [Ca24]: for $C\ge1$ and every sufficiently large $k$ there exist integers $0<x<y$ with $y>x+k$ such that $\operatorname{lcm}\{x,\ldots,x+k-1\}>C\cdot\operatorname{lcm}\{y,\ldots,y+k\}$. Translation to the site's notation (authored here, one line): $\operatorname{lcm}\{x,\ldots,x+k-1\}=M(x-1,k)$ and $\operatorname{lcm}\{y,\ldots,y+k\}=M(y-1,k+1)$, so with $n=x-1$, $m=y-1$ the theorem gives $M(n,k)>C\cdot M(m,k+1)$, and $y>x+k$ is $m\ge n+k+1$, which implies $m\ge n+k$; taking $C=1$, every sufficiently large $k$, in particular infinitely many $k\ge3$, yields a triple $(m,n,k)$ as asked. The paper's minimal examples $\operatorname{lcm}\{53,\ldots,59\}>\operatorname{lcm}\{63,\ldots,70\}$ and $\operatorname{lcm}\{37,\ldots,44\}>\operatorname{lcm}\{48,\ldots,56\}$ are $M(52,7)>M(62,8)$ and $M(36,8)>M(47,9)$ (recomputed here). The proof (pp. 2--4) has this structure: with $M=\operatorname{lcm}\{1,\ldots,k\}$, Chinese-remainder families $x_{\vec a}\equiv1\pmod m$, $y_{\vec b}\equiv0\pmod m$ with prescribed residues modulo the primes in $(\sqrt k,k]$, a counting claim (Claim 4) and the density of primes near $k/2$ and $k$ place $y$ in $(\frac M{5C}(1+\frac1k),\frac M{4C}-k)$ and $x$ just below $y$ with $y-x>k$, and the valuation identity (Claim 5) $y\cdots(y+k)/\operatorname{lcm}=M\cdot x\cdots(x+k-1)/\operatorname{lcm}$ gives the ratio $\ge\frac M{y+k}(\frac xy)^k>4C(\frac k{k+1})^k>C$. Read depth: the statements of Theorem 1, Conjecture 2, Question 3 and Claims 4 and 5 are checked; the proof is not checked step by step and nothing here is independently reviewed; the argument is a candidate for an independent whole-argument review. Acceptance evidence: the site's label and commentary (11 January 2026), the thread's report of the formalization, and the formal-conjectures collection's research solved category; no journal record (the arXiv listing has no journal reference, a Crossref bibliographic query for the title returned nothing, and Semantic Scholar lists no citing paper on 2026-09-18), so the preprint qualification applies until a refereed version or an independent review appears. The paper's Conjecture 2 (the larger block with $C$ extra elements) and Question 3 (a Chinese-remainder density statement implying it) remain open per the paper.

Formalization and the Lean label. The site's "(Lean)" suffix is a catalog label. The formal-conjectures file's three formal_proof attributes name src/latest/ErdosProblems/Erdos678.lean of plby/lean-proofs at its commit of 1 August 2026, the version the claim page links (the repository's head on 2026-09-18 dated from 2026-09-15); that version of the file has 179,114 bytes and 2,536 lines. Its header records Lean v4.32.0 and Mathlib v4.32.0, names Cambie as informal author and Aristotle and Boris Alexeev as formal authors, and says it formalizes "the first main result" of the paper, proving MainTheoremStatement (Theorem 1) "assuming the DensityHypothesis (which follows from results on prime gaps, e.g., Baker-Harman-Pintz)". It imports Mathlib and PrimeNumberTheoremAnd.Consequences (an external Lean project), proves main_theorem (h_density : DensityHypothesis) : MainTheoremStatement, derives main_theorem_given_pnt : MainTheoremStatement from that project, and ends with main_theorem_expanded, erdos_678 (the site's statement), erdos_678_kmn_infinite (infinitely many triples), not_erdos_678_fc and not_erdos_678_other (the fixed-$k$ readings are false, proved without the prime number theorem); its text contains no sorry, no axiom declaration and no native_decide, and its closing #print axioms comments list propext, Classical.choice and Quot.sound for each end result. The repository's index page for the file lists builds for five Mathlib versions. The version the thread announced on 8 January 2026 (a different file of the same repository) declared pi_alt, a prime number theorem statement, as an axiom; the pinned version derives the density hypothesis from the imported project instead. Nothing was built or kernel-checked here, the imported project's own axioms were not examined, and no local kernel credit is claimed. The community database records formal_status Lean since 7 January 2026 and no formal-proof URL.

Adjacent questions from the source (not this problem). $n_k$, the least $n$ with $M(n,k)>M(m,k)$ for some $m\ge n+k$: $n_k/k\to\infty$ per Erdős, no upper bound on record; $u_k$: Erdős prints $u_k=(1+o(1))k$ and $u_k>k$ as easy, and the site's commentary records the many counterexamples to $M(t,k)\le M(T,k)$ for $t<\min(u_k,T)$. The equality conjecture $M(n,k)\ne M(m,k)$ is Problem 677.

Search scope. None of the routes below found a refereed version of [Ca24], an independent review of its proof, a dispute, or a copy of [Er92e].

• The site: problem page, discussion thread and proof-claim tab; formal-conjectures 678.lean at the pinned commit; the community database (2026-09-18); the Lean file, its commit and the repository's head and index pages through the GitHub API.
• arXiv: the API record of 2410.09138 (v1 only; no journal reference) and the queries abs:"least common multiple" AND abs:consecutive AND abs:Erdős (one record, this paper) and abs:"least common multiple" AND abs:"consecutive integers" (none).
• Crossref: a bibliographic query for the paper's title (no record).
• Semantic Scholar: the citation list of arXiv:2410.09138 (no records).
• The publishing society's site for [Er92e]: archim.org.uk/eureka/archive/ and archim.org.uk/eureka/ (both HTTP 404; the archive's current location was not found).
• The primary sources: [Ca24] pp. 1--5 and [Er79] pp. 67--68.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [Er92e].

Remaining gaps. (1) The status rests on a preprint with no refereed publication and no independent expert review; a refereed version or a whole-argument review is the reopening condition for the qualification. (2) The Lean development is not built here and rests on an imported external project for the prime number theorem. (3) [Er92e] is not held; the site's account of its wording and of the referee's identity is second-hand.

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.

• cambie_2024_resolution_erdos_problem_least_common_multiples
• cambie_2024_resolution_erdos_problem_least_common_multiples / theorem_1
• erdos_1979_unconventional_problems_number_theory_math_mag