Cambie 2024 resolution erdos problem least common multiples

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theorem_1: The theorem resolving Problem 678 in a strong form: for every constant C and all large k, some block of k consecutive integers has a least common multiple more than C times that of a later block of k+1.

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Stijn Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).

The retained folder-name PDF is arXiv:2410.09138v1 (stamped "11 Oct 2024"; the text is dated October 15, 2024), five pages, the only version on arXiv on 2026-09-18. No journal record was found on that date: the arXiv listing carries no journal reference or DOI, a Crossref bibliographic query for the title returned no record, and Semantic Scholar lists no citing paper. Read status: claims checked for Theorem 1, Conjecture 2, Question 3 and the statements of Claims 4 and 5 (text layer; pp. 1--2 also on the page images); the two-page proof of Theorem 1 was read for its structure and not checked step by step; nothing here is independently reviewed. The arXiv record (https://arxiv.org/abs/2410.09138, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Theorem 1 proves that for any constant C >= 1 and every sufficiently large k there are integers 0 < x < y with y > x + k such that lcm{x,...,x+k-1} > C·lcm{y,...,y+k}, so the ratio of the two least common multiples can be made arbitrarily large. This answers affirmatively Erdős's 1979 question, recorded as problem 678, of whether infinitely many such configurations exist, and in a strong quantitative form. The proof is elementary: writing M = lcm{1,...,k} = m · prod_{sqrt(k) < p <= k} p, it builds via the Chinese remainder theorem large families of residue vectors whose solutions x and y force many primes to divide the small block and to be wasted on the large block. The paper records small explicit examples, such as lcm{53,...,59} > lcm{63,...,70} and lcm{37,...,44} > lcm{48,...,56}, and states Conjecture 2, a stronger version in which the larger set has C extra elements, which the author reduces (Subsection 2.1) to a density statement for Chinese-remainder solutions, formulated as Question 3.

Source: https://arxiv.org/abs/2410.09138.

Bears on. #678: Theorem 1 with the substitution n = x-1, m = y-1 is the site's statement M(n,k) > M(m,k+1) with m >= n+k, for every large k and hence for infinitely many triples; the site's label PROVED (LEAN) rests on this preprint and on an external Lean formalization of it.

Results to transcribe.

• Theorem 1 (p. 2): For every C >= 1 and all large k there exist 0 < x < y with y > x+k and lcm{x,...,x+k-1} > C·lcm{y,...,y+k}.
• Conjecture 2 (p. 2): For every constant C there are k and 0 < x < y with y > x+k such that lcm{x,...,x+k-1} > lcm{y,...,y+k+C-1}.
• Question 3 (p. 2): A Chinese-remainder density statement about residues in initial intervals modulo the primes between sqrt(k) and k, which would imply Conjecture 2 (Subsection 2.1, p. 4).