A binomial product representation for p-adic numbers (Q1825900)

Let \({\mathbb{Z}}_ p\) denote the ring of p-adic integers, p a prime number. The p-adic numbers A lying in the open unit ball \(1+p{\mathbb{Z}}_ p\) with centre 1 have p-adic expansions of standard type \(A=1+\sum^{\infty}_{n=1}c_ np^ n\), \(0\leq c_ n\leq p-1\). In this note the authors show by an elementary method that these numbers have another unique expansion, namely an infinite product representation of the form \(A=\prod^{\infty}_{n=1}(1+p^{r_ n})^{b_ n}\), with \(1\leq b_ n\leq p-1\), \(r_ n\in {\mathbb{N}}\), \(r_{n+1}>r_ n\). Also the question when A is a rational integer is considered.