Grimm's Conjecture

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.

Let $n+1,\ldots,n+k$ be composite positive integers. The proposed assignment consists of pairwise distinct primes $q_1,\ldots,q_k$ with $q_i\mid n+i$ for each $i$. A number may have several prime factors, but only one is selected for its assigned representative. The assertion quantifies over every finite run of consecutive composites. [S1] [S2]

Statement recorded in the supplied source.

Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove Hall's matching condition for every composite block or produce an explicit failing block. [S1]

Short English statement

Can the numbers in any consecutive run of composites each be assigned a prime divisor without assigning the same prime twice?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1240 (NT-033), 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 S. Laishram and T. N. Shorey, Grimm's conjecture on consecutive integers, International Journal of Number Theory 2 (2006), 207-211. https://doi.org/10.1142/S1793042106000498. Bibliographic information from the supplied record.

• S3 R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Problem B32, Springer (2004). https://doi.org/10.1007/978-0-387-26677-0. Bibliographic information from the supplied record.