Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 4
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Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $y $ be a rational $p$-adic integer.
Let $\ldots d_n \ldots d_2 d_1 d_0$ be the canonical expansion of $y$.
Let:
- $y = \dfrac a b : a \in \Z, b \in Z_{> 0}$
Let:
- $\forall n \in \N: \exists A_n, r_n \in \Z$:
- $(\text a) \quad \dfrac a b = A_n + p^{n + 1} \dfrac {r_n} b$
- $(\text b) \quad 0 \le A_n \le p^{n + 1} - 1$
- $(\text c) \quad \ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$
Then:
- $\forall n \in \N: r_n = d_{n + 1} b + p r_{n + 1}$
Proof
We have:
\(\ds \forall n \in \N: \, \) | \(\ds \dfrac a b\) | \(=\) | \(\ds A_n + p^{n + 1} \dfrac {r_n} b\) | by hypothesis | ||||||||||
\(\ds \) | \(=\) | \(\ds A_{n + 1} + p^{n + 2} \dfrac {r_{n + 1} } b\) | by hypothesis | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N: \, \) | \(\ds A_{n + 1} + p^{n + 2} \dfrac {r_{n + 1} } b\) | \(=\) | \(\ds A_n + p^{n + 1} \dfrac {r_n} b\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N: \, \) | \(\ds A_{n + 1} - A_n\) | \(=\) | \(\ds p^{n + 1} \paren {\dfrac {r_n - p r_{n + 1} } b}\) | rearranging terms | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N: \, \) | \(\ds b \paren {A_{n + 1} - A_n}\) | \(=\) | \(\ds p^{n + 1} \paren {r_n - p r_{n + 1} }\) | rearranging terms | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N: \, \) | \(\ds b\) | \(\divides\) | \(\ds p^{n + 1} \paren {r_n - p r_{n + 1} }\) |
From Characterization of Rational P-adic Integer:
- $p \nmid b$
From Prime not Divisor implies Coprime:
- $b, p$ are coprime
From Integer Coprime to all Factors is Coprime to Whole:
- $b, p^{n+1}$ are coprime
Hence: $b \nmid p^{n + 1}$
From Euclid's Lemma:
- $b \divides \paren {r_n - p r_{n + 1} }$
Then:
- $\dfrac {r_n - p r_{n + 1} } b \in \Z$
Hence:
- $A_{n + 1} \equiv A_n \pmod {p^{n + 1} }$
By definition of coherent sequence:
- the sequence $\sequence{A_n}$ is a coherent sequence
From Coherent Sequence is Partial Sum of P-adic Expansion:
- the sequence $\sequence{A_n}$ is the sequence of partial sums of a $p$-adic expansion $\ds \sum_{i \mathop = 0}^n e_i p^i$
Hence:
- $y = \dfrac a b = \ds \sum_{n \mathop = 0}^\infty e_n p^n$
From P-adic Number is Limit of Unique P-adic Expansion:
- $\forall n \in \N: e_n = d_n$
By definition of partial sums:
- $\forall n \in \N: A_{n + 1} = A_n + d_{n + 1} p^{n + 1}$
Hence:
\(\ds \forall n \in \N: \, \) | \(\ds d_{n + 1} p^{n + 1}\) | \(=\) | \(\ds A_{n + 1} - A_n\) | |||||||||||
\(\ds \) | \(=\) | \(\ds p^{n + 1} \paren {\dfrac {r_n - p r_{n + 1} } b}\) | a priori | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N: \, \) | \(\ds d_{n + 1}\) | \(=\) | \(\ds \paren {\dfrac {r_n - p r_{n + 1} } b}\) | dividing both sides by $p^{n + 1}$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \N: \, \) | \(\ds r_n\) | \(=\) | \(\ds d_{n + 1} b + p r_{n + 1}\) | rearranging terms |
$\blacksquare$