Resumen de Catalan's intriguing factorial problem

Thomas Koshy

• This article investigates the numbers , originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers Cn, we develop some divisibility properties for Cn. In particular, we establish that , where fkdenotes the kth Fermat number and Mkthe kth Mersenne number 2k?-?1. Finally, we develop an explicit formula for Xm,nusing Pascal's triangle and Catalan numbers, and extract several interesting byproducts from it.