Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 10
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Theorem
Let $p$ be a prime.
Let $b \in Z_{>0}$ be a (strictly) positive integer such that $b, p$ are coprime.
Let $\sequence {d_n}$ be a sequence of $p$-adic digits.
Let $\sequence {r_n}$ be an integer sequence such that:
| \(\text {(1)}: \quad\) | \(\ds \forall n \in \N: \, \) | \(\ds r_n\) | \(=\) | \(\ds d_{n + 1} b + p r_{n + 1}\) | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds \exists n_0 \in \N: \forall n \ge n_0: \, \) | \(\ds -b\) | \(\le\) | \(\ds r_n \le 0\) |
Let:
- $n, k \in \N : k > 0 : r_n = r_{n + k}$
Then:
- $d_{n + 1} = d_{n + k + 1}$
- $r_{n + 1} = r_{n + k + 1}$
Proof
We have:
| \(\ds d_{n + 1} b + p r_{n + 1}\) | \(=\) | \(\ds r_n\) | by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds r_{n + k}\) | by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds d_{n + k + 1} b + p r_{n + k + 1}\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds p \paren{ r_{n + k + 1} - r_{n + 1} }\) | \(=\) | \(\ds \paren {d_{n + 1} - d_{n + k + 1} }b\) | re-arranging terms |
As $b, p$ are coprime:
- $p \nmid b$
From Euclid's Lemma:
- $p \divides \paren {d_{n + 1} - d_{n + k + 1} }$
By definition of $p$-adic digits:
- $d_{n + 1}, d_{n + k + 1} \in \set {0, 1, \ldots, p - 1}$
Hence:
- $d_{n + 1} = d_{n + k + 1}$
We have:
| \(\ds p \paren{ r_{n + k + 1} - r_{n + 1} }\) | \(=\) | \(\ds \paren {d_{n + 1} - d_{n + k + 1} } b\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | as $d_{n + 1} = d_{n + k + 1}$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {r_{n + k + 1} - r_{n + 1} }\) | \(=\) | \(\ds 0\) | as $p \ne 0$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds r_{n + 1}\) | \(=\) | \(\ds r_{n + k + 1}\) |
$\blacksquare$