User:Leigh.Samphier/Sandbox/Definition:Eventually Periodic P-adic Expansion

Definition
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $x \in \Q_p$.

Let $\ldots d_n \ldots d_2 d_1 d_0. d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.

Let there be a finite sequence of $k$ digits of $x$:
 * $\tuple {d_{r + k - 1} \ldots d_{r+1} d_r }$

such that $r \ge 0$ and for all $n \in \Z_{\ge 0}$ and for all $j \in \set {0, 2, \ldots, k - 1}$:
 * $d_{r + j + n k} = d_{r + j}$

where $k$ is the smallest $k$ to have this property.

That is, let the canonical expansion of $x$ be of the form:


 * $\ldots d_{r + k - 1} \ldots d_{r+1} d_r d_{r + k - 1} \ldots d_{r+1} d_r d_{r + k - 1} \ldots d_{r+1} d_r \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$

That is, $\tuple {d_{r + k - 1} \ldots d_{r+1} d_r }$ repeats.

Then the canonical expansion of $x$ is said to be eventually periodic.

Also see

 * User:Leigh.Samphier/Sandbox/Definition:Periodic P-adic Expansion


 * User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic