Problem 889

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Statement. 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 $n\to \infty$?

Status. Open.

Source. erdosproblems.com/889, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #889, https://www.erdosproblems.com/889.

References.

• [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428-430.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B27 "The number of prime factors of $n+k$ which don't divide $n+i$, $0\le i<k$", p. 126, states the question $v_0(n)\to\infty$ and the exceptions $n=1,2,3,4,7,8,16$ to $v_0(n)>1$. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1967_problems_prime_factors_consecutive_integers
• erdos_1967_problems_prime_factors_consecutive_integers / theorem_p428
• guy_2004_unsolved_problems_number_theory