Solving equations in \(\beta\mathbb{N}\) (Q1590065)

The familiar operations of \(+\) and \(\cdot\) on the positive integers \(\mathbb{N}\) extend uniquely to semigroup operations on the Stone-Čech compactification \(\beta\mathbb{N}\) in such a way that all right translations are continuous and left translations by members of the embedded copy of \(\mathbb{N}\) are continuous. The author establishes various results showing that the set of solutions in \(\beta\mathbb{N}\) to various arithmetic equations involving both the extended \(+\) and \(\cdot\) is in general quite restricted. For example for \(p\in {\mathbb{N}}^*=\beta\mathbb{N}\setminus \mathbb{N}\), \(a,b\in\mathbb{N}\), and \(u,v\in\beta\mathbb{N}\), if \(u+ap=v+bp\), then \(a=b\). The author also establishes other related types of results.