Characterisation of Terminal P-adic Expansion/Sufficient Condition
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Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $a \in \N$.
Let $k \in \Z$.
Let $x = \dfrac a {p^k}$.
Then:
- the $p$-adic expansion of $x$ terminates
Proof
From Basis Representation Theorem, $a$ can be expressed uniquely in the form:
- $\ds a = \sum_{j \mathop = 0}^n d_j p^j$
where:
- $n$ is such that $p^n \le a < p^{n + 1}$
- all the $d_j$ are such that $0 \le d_j < p$.
We have:
\(\ds x\) | \(=\) | \(\ds \dfrac a {p^k}\) | Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\ds \sum_{j \mathop = 0}^n d_j p^j} {p^k}\) | Replacing $a$ with base $p$ expression | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^n d_j p^{j - k}\) | Dividing each term by $p^k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = -k}^{n-k} d_{i+ k} p^i\) | Re-indexing with $i = j - k$ |
Let:
- $m = \begin{cases}
-k & : -k \le 0\\ 0 & : 0 < -k \end{cases}$
For each $i : m \le i \le n-k$, let:
- $e_i = \begin{cases}
d_{i + k} & : -k \le i \le n-k\\ 0 & : m \le i < -k \end{cases}$
For each $i > n-k$, let:
- $e_i = 0$
Then:
- $x = \ds \sum_{i \mathop = m}^\infty e_i p^i$
where:
- $\forall i \ge m: 0 \le e_i < p$
- $\forall i > n-k: e_i = 0$
Hence $\ds \sum_{i \mathop = m}^\infty e_i p^i$ is a terminal $p$-adic expansion by definition.
From P-adic Expansion Representative of P-adic Number is Unique, the $p$-adic expansion of $x$ is:
- $x = \ds \sum_{i \mathop = m}^\infty e_i p^i$
$\blacksquare$