Erdős Problem #679

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 lower cutoff on $k$ is chosen using $\epsilon$ alone, before the infinitely many endpoints $n$ are selected. In particular, it cannot be moved close to $n$ to discard bad values. The second question asks about failure of a stronger uniform bound.

For $n=\prod_pp^{v_p(n)}$, $\omega(n)=\#\{p:v_p(n)>0\}$ counts distinct prime factors, whereas $\Omega(n)=\sum_pv_p(n)$ would count with multiplicity. The displayed question uses the former. [S1] [S2]

Statement recorded in the supplied source.

Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that $ \omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k} $ for all $k<n$ which are sufficiently large depending on $\epsilon$ only? Can one show the stronger version with $ \omega(n-k) < \frac{\log k}{\log\log k}+O(1) $ is false? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove the original (1+epsilon)log k/log log k simultaneous upper bound for infinitely many n or establish Lau's predicted logarithmic obstruction. [S1]

Short English statement

Are there infinitely many endpoints behind which every sufficiently distant integer has no more than the prescribed near-logarithmic number of prime factors?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2250 (EP-679), 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 679, https://www.erdosproblems.com/679. 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.