Let $k\geq 3$. Are there infinitely many $m,n$ with $m\geq n+k$ such that \[M(n,k)>M(m,k+1)?\]

OPEN

Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$.

Let $k\geq 3$. Are there infinitely many $m,n$ with $m\geq n+k$ such that \[M(n,k)>M(m,k+1)?\]

It is easy to see that there are infinitely many solutions to $M(n,k)>M(m,k)$. The referee of [Er79] found $M(96,7)>M(104,8)$ and $M(132,7)>M(139,8)$.

If $n_k$ is the smallest $n$ with this property (for some $m$) then are there good bounds for $n_k$? Erdős writes that he could prove $n_k/k\to \infty$, but knew of no good upper bounds.

If $u_k$ is minimal such that $M(u_k,k)>M(u_k+1,k)$ and $t<\min(u_k,T)$ then is it true that $M(t,k)\leq M(T,k)$?

See also [677].