Problem 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 $n\to \infty$?
Is it true that\[v_0(n)=\max_{k\geq 0}v(n,k)\to \infty\]as $n\to \infty$?
Categories:
Number Theory
Progress
A question of Erdős and Selfridge [ErSe67], who could only show that $v_0(n)\geq 2$ for $n\geq 17$. More generally, they conjecture that\[v_l(n)=\max_{k\geq l}v(n,k)\to \infty\]as $n\to \infty$, for every fixed $l$, but could not even prove that $v_1(n)\geq 2$ for all large $n$.This is problem B27 of Guy's collection [Gu04].
Source: erdosproblems.com/889 | Last verified: January 19, 2026