Coherent Sequence is Partial Sum of P-adic Expansion

From ProofWiki
Jump to navigation Jump to search

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