Erdős Problem #677

declared status 'open'. Formalized: yes. Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$. Is it true that for all $m\geq n+k$$$M(n,k) \neq M(m,k)?$$ Current best: The Thue-Siegel theorem implies that, for fixed $k$, there are only finitely many $m,n$ such that $m\geq n+k$ and $M(n,k)=M(m,k)$. In general, how many solutions does $M(n,k)=M(m,l)$ have when $m\geq n+k$ and $l>1$? Erd\H{o}s expects very few (and none when $l\geq k$). In [Er79d] Erd\H{o}s conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\geq n+k$ then $\prod_{i\leq k}(n+i)$ and $\prod_{i\leq k}(m+i)$ cannot have the same set of prime factors. Prize: no. Tags: number theory.

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