Erdős Problem #889

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.

Prime factors are counted distinctly. A prime dividing $n+k$ occurred among the preceding $k$ integers exactly when it is at most $k$. Hence $v(n,k)=\#\{p:p\mid n+k,\ p>k\}$, with no preceding integers when $k=0$. [S1] [S2]

Statement recorded in the supplied source.

For $k\geq 0$ and $n\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i<k$. Equivalently, $v(n,k)$ counts the number of prime factors of $n+k$ which are $>k$. Is it true that $ v_0(n)=\max_{k\geq 0}v(n,k)\to \infty $ as $n\to \infty$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove v_0(n) tends to infinity. [S1]

Short English statement

Starting from larger integers, must some later term introduce arbitrarily many prime factors absent from every preceding term of that block?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2368 (EP-889), 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 889, https://www.erdosproblems.com/889. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430. Bibliographic information from the supplied record.

• S4 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.