Problem 677

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

Is it true that for all $m\geq n+k$

M(n,k) \neq M(m,k)?

Source: erdosproblems.com/677

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label. No proof or disproof of the same- $k$ conjecture was found in the search whose scope the Current assessment records. What is on record: the remark, printed in the 1980 monograph (p. 76) and repeated by the site, that the Thue–Siegel theorem gives, for each fixed $k$ , only finitely many pairs $m\ge n+k$ with $M(n,k)=M(m,k)$ (stated there without proof or reference); Erdős's two solutions of the general equation $M(n,k)=M(m,l)$ with $l\ne k$ , $M(4,3)=M(13,2)=210$ and $M(3,4)=M(19,2)=420$ , and further solutions posted in the thread in 2026, $M(8,4)=M(43,2)=1980$ , $M(153,3)=M(1363,2)=1861860$ and the padded forms $M(n,7-n)=M(19,2)$ for $n\le2$ (all recomputed here); Erdős's stronger question of 1979 ([Er79d]), whether two disjoint blocks of $k>2$ consecutive integers can have products with the same set of prime factors more than finitely often; and a forum repository of prover-generated Lean lemmas, a lead. This is a bounded negative finding, not a certificate of openness.

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: Erdős 1979.
Er79d: Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. 33 (1979), 71--80; Section 3, printed p. 78: the same-length conjecture, the same-prime-factors question and the ratio $\alpha(m,n,k)$ . Library home: Erdos 1979.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28 (1980); printed p. 76: the same-length conjecture with the Thue–Siegel finiteness remark; the site gives no page. Library home: Erdős and Graham 1980 (the card also locates the product question on printed p. 74).
Gu04: Guy, R. K., Unsolved Problems in Number Theory, 3rd ed. Problem Books in Mathematics, Springer (2004), xviii+437 pp. B35 "Products of consecutive numbers with the same prime factors", printed p. 138: with $L(n;k)$ the l.c.m. of $n+1,\ldots,n+k$ , "Erdős conjectures that for $l>1$ , $n\ge m+k$ , $L(m;k)=L(n;l)$ has only a finite number of solutions", the examples $L(4;3)=L(13;2)$ and $L(3;4)=L(19;2)$ , and the questions on $L(n;k)>L(n-k;k)$ , citing Erdős's 1980 Monthly note; no proofs. Library home: Guy 2004.
Ca24: Cambie, S., Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138v1 (2024); adjacent context only (Problem 678's inequality, not this equation). Library home: Cambie 2024.
FaKa09: Farhi, B. and Kane, D., New results on the least common multiple of consecutive integers. Proc. Amer. Math. Soc. 137 (2009), no. 6, 1933--1939, DOI 10.1090/S0002-9939-08-09730-X (the thread links the authors' preprint at cseweb.ucsd.edu); on the exact period of $n(n+1)\cdots(n+k)/\operatorname{lcm}(n,\ldots,n+k)$ ; linked from the thread as possibly of interest; context, not this problem; not held.

Problem 677

Status

Proof claims

Discussion