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$p$-adic logarithm of integers
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Numbers
$p$
$k$ 
$\log_p(k)$
INPUT{numbers.yaml} (not shown in preview)
Definition
This table contains $p$-adic logarithms $\log_p(k) \in \mathbb{Z}_p$ of certain integers $k$.
Parameters
$p$
—   integer (prime)
$k$
—   integer ($k = 1 \mod p$)
Comments
(1)
The $p$-adic logarithm is defined by its Taylor series for all $x \in 1 + p\mathbb{Z}_p$. It extends via its functional equation to all of $\mathbb{Z}_p^\times$: Any element of $\mathbb{Z}_p^\times$ can be uniquely written as a product $ax$ of some $x \in 1 + p\mathbb{Z}_p$ and some Teichmüller representative $a \in \mathbb{Z}_p$. As $a$ is a root of unity, the functional equation for $\log_p$ dictates $\log_p(a)=0$ and thus $\log_p(ax) = \log_p(x)$. Similarly, we can extend $\log_p$ to any $x\in \mathbb{Q}_p^\times$ by prescribing an arbitrary value for $\log_p p$, such as $\log_p p = 0$.
Links
Data properties
Entries are of type: p-adic number
Table is complete: no