Coherent Sequence is Partial Sum of P-adic Expansion
Theorem
Let $p$ be a prime number.
Let $\sequence {\alpha_n}$ be a coherent sequence.
Then there exists a unique $p$-adic expansion of the form:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
- $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$
Informal Proof
Consider the $\ds \alpha_n$ written in base $p$.
To reduce an integer modulo $p^n$, it is simply a matter of stripping off all but the last $n$ digits.
So the coherence condition:
- $\alpha_{n + 1} \equiv \alpha_n \mod{p^{n+1}}$
means that the last $n + 1$ digits of both integers are the same.
So the sequence $\sequence {\alpha_n}$ can be written:
\(\ds \alpha_0\) | \(=\) | \(\ds d_0\) | $0 \le d_0 \le p - 1$ | |||||||||||
\(\ds \alpha_1\) | \(=\) | \(\ds d_0 + d_1 p\) | $0 \le d_1 \le p - 1$ | |||||||||||
\(\ds \alpha_2\) | \(=\) | \(\ds d_0 + d_1 p + d_2 p^2\) | $0 \le d_2 \le p - 1$ | |||||||||||
\(\ds \alpha_3\) | \(=\) | \(\ds d_0 + d_1 p + d_2 p^2 + d_3 p^3\) | $0 \le d_3 \le p - 1$ | |||||||||||
\(\ds \ldots\) | \(\) | \(\ds \) | $0 \le d_n \le p - 1$ |
Putting this together, we get the $p$-adic expansion:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
- $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$
$\blacksquare$
Proof
By definition of a coherent sequence:
- $\forall n \in \N: 0 \le \alpha_n < p^{n + 1}$
From Zero Padded Basis Representation, for all $n \in \N$ there exists a sequence $\sequence {b_{j, n} }_{0 \le j \le n} :$
- $(1) \quad \ds \alpha_n = \sum_{j \mathop = 0}^n b_{j, n} p^j$
- $(2) \quad \forall j \in \closedint 0 n : 0 \le b_{j, n} < p$
Lemma
- $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n b_{i, i} p^i$
$\Box$
For all $n \in \N$, let:
- $d_n = b_{n, n}$
Then:
- $\forall n \in \N : 0 \le d_n < p$
By definition:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
is a $p$-adic expansion.
From the lemma:
- $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$