Products of Catalan numbers which are squares

Abstract

Let \(C_{n}\) be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each \(k\ge 3\), there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of \(x\in \mathbb {N}_{+}\) such that \(C_{x}C_{x+1}\) is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.