Coherent Sequence is Partial Sum of P-adic Expansion/Informal Proof
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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:
- $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$