Erdős Problem #826

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.

The meaning is that some absolute $C$ and infinitely many starting integers $n$ satisfy $\tau(n+k)\leq Ck$ for every positive $k$. Choosing a separate growing constant for each shift would not express the stated uniform estimate.

The divisor-count function is $\tau(n)=\#\{d\in{\mathbb{N}}:d\mid n\}$. Divisors are positive, including one and $n$; the function is unrelated to a graph transversal number using the same letter elsewhere. [S1] [S2]

Statement recorded in the supplied source.

Are there infinitely many $n$ such that, for all $k\geq 1$, $ \tau(n+k)\ll k? $ [S1]

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

Short English statement

Can infinitely many starting integers have every later divisor count bounded by one universal constant times its distance from the starting point?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2325 (EP-826), 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 826, https://www.erdosproblems.com/826. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Cheuk Fung Lau, On the Number of Prime Factors of Consecutive Integers, arXiv:2604.15042 (2026). https://arxiv.org/abs/2604.15042. Bibliographic information from the supplied record.