# Representation of p-adic integers using primitive roots

#### SlipEternal

MHF Helper
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?