Let p be an odd prime and let r be any positive integer that is a primitive root module p^2. Let X_p = \varprojlim \Bbb{Z} / (p-1)p^n \Bbb{Z} be the inverse limit of the rings \Bbb{Z} / (p-1)p^n \Bbb{Z}. This is useful because for any unit u \in \Bbb{Z}/p^n\Bbb{Z}, there exists K \in \Bbb{Z} / (p-1)p^{n-1}\Bbb{Z} with r^k \equiv u \pmod{p^n} for all k \equiv K \pmod{(p-1)p^{n-1}}. So, any element of the p-adic integers: x \in \Bbb{Z}_p, you can represent it as x = p^a r^b for some a \in \mathbb{N}, b\in X_p. I haven't seen this examined at all. Multiplication of terms in \Bbb{Z}_p would just correlate to addition of in X_p. There is no operation in X_p that correctly models addition in \Bbb{Z}_p. But, exponentiation in \Bbb{Z}_p closely correlates to multiplication in X_p. Has anyone seen any examination of this type of representation for p-adic integers?