Remark on factorials that are products of factorials
Let \(1<a\leq b<n\) be integers satisfying \(n!=a!b!\). Erdős proved in 1993 that \(n-b<5\log\log n\) for all large enough \(n\). In the paper under review the authors improve this by showing that if \(n!=a_1!a_2!\cdots a_k! b!\), where \(1<a_1\leq a_2\leq \cdots \leq a_k<b<n\), then \(n-b<(1/\log 2+\varepsilon)\log\log n\) holds for all \(n>n_{\varepsilon}\) sufficiently large, where \(\varepsilon>0\) is arbitrary. The proof uses the formula of the exponent of \(2\) in the involved factorials, as well as the celebrated result of \textit{R. C. Baker, G. Harman} and \textit{J. Pintz} concerning the distance between consecutive primes [Proc. Lond. Math. Soc., III. Ser. 83, No. 3, 532--562 (2001; Zbl 1016.11037)]. Conditionally under the \(abc\)-conjecture, the reviewer [Math. Proc. Camb. Philos. Soc. 143, No. 3, 533--542 (2007; Zbl 1132.11017)], has shown that under the above hypotheses one has \(n-b=1\) for all large enough \(n\). Erdős's paper mentioned in the text does not appear in the list of references, which are not listed alphabetically.
scientific article; zbMATH DE number 5980673