Remark on Factorials that are Products of Factorials

In a paper published in 1993, Erdős proved that if $n!=a!b!$, where $1<a\le b$, then the difference between $n$ and $b$ does not exceed $5\log\log n$ for large enough $n$. In the present paper, we improve this upper bound to $((1+\epsilon)/\log 2)\log\log n$ and generalize it to the equation $a_1!a_2!\dots a_k!=n!$. In a recent paper, F. Luca proved that $n-b=1$ for large enough $n$ provided that the ABC-hypothesis holds.