Problem #686: Can every integer $N\geq 2$ be written...

Can every integer $N\geq 2$ be written as\[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\]for some $k\geq 2$ and $m\geq n+k$?

SOLVED

Problem Statement

Can every integer $N\geq 2$ be written as\[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\]for some $k\geq 2$ and $m\geq n+k$?

Categories: Number Theory

If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?

See also [677].

Source: erdosproblems.com/686 | Last verified: January 16, 2026