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