Problem

Source: Romania TST 2015 Day 3 Problem 3

Tags: least common multiple, function, Romanian TST, number theory, RIP



If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$.Let $f(n)$ be the largest positive integer $ k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that : (a) $f(n)<3\sqrt{n}$ for all positive integers $n$ . (b) If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$.