Erdős Problem #677

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For positive integers, $[a_1,\ldots,a_j]=\operatorname{lcm}(a_1,\ldots,a_j)$ is the least positive common multiple, and $(a,b)=\gcd(a,b)$ is their greatest common divisor. Brackets used this way are not interval notation. [S1] [S2]

Statement recorded in the supplied source.

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)? $ [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

Can two disjoint blocks of consecutive positive integers of the same length ever have identical least common multiples?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2249 (EP-677), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 677, https://www.erdosproblems.com/677. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine 52 (1979), 67-70. https://combinatorica.hu/~p_erdos/1979-22.pdf. Bibliographic information from the supplied record.

• S4 Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. Bibliographic information from the supplied record.

• S5 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.