Problem 889

Erdős Problem #889: declared status 'open'. Formalized: yes. 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 $n\to \infty$? Current best: A question of Erd\H{o}s 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 $n$. Prize: no. Tags: number theory.