# (wrong string) ée

From: <git_AT_suckless.org>
Date: Mon, 25 Jul 2016 16:57:13 +0200 (CEST)

commit 019da3a9e7f81cd882d0383ac707ce098013b4a9
Author: Mattias Andrée <maandree_AT_kth.se>
AuthorDate: Mon Jul 25 16:38:43 2016 +0200
Commit: Mattias Andrée <maandree_AT_kth.se>
CommitDate: Mon Jul 25 16:38:43 2016 +0200

Manual: The Kronecker symbol

Signed-off-by: Mattias Andrée <maandree_AT_kth.se>

diff --git a/doc/not-implemented.tex b/doc/not-implemented.tex
index 6cff52e..ed912ba 100644
--- a/doc/not-implemented.tex
+++ b/doc/not-implemented.tex
_AT_@ -163,7 +163,8 @@ so a compressed lookup table can be used for small $p$.
\left ( \frac{a}{n} \right ) =
\prod_k \left ( \frac{a}{p_k} \right )^{n_k},
}\)
-where $n$ = $\displaystyle{\prod_k p_k^{n_k}}$, and $p_k \in \textbf{P}$.
+where $\displaystyle{n = \prod_k p_k^{n_k} > 0}$,
+and $p_k \in \textbf{P}$.
\vspace{1em}

Like the Legendre symbol, the Jacobi symbol is $n$-period over $a$.
_AT_@ -197,14 +198,65 @@ Use the following algorithm to calculate the Jacobi symbol:
\STATE \textbf{start over}
\end{algorithmic}
\end{minipage}
-\vspace{1em}
-

\subsection{Kronecker symbol}
\label{sec:Kronecker symbol}

-TODO
+The Kronecker symbol
+$\displaystyle{\left ( \frac{a}{n} \right )}$
+is a generalisation of the Jacobi symbol,
+where $n$ can be any integer. For positive
+odd $n$, the Kronecker symbol is equal to
+the Jacobi symbol. For even $n$, the
+Kronecker symbol is $2n$-periodic over $a$,
+the Kronecker symbol is zero for all
+$(a, n)$ with both $a$ and $n$ are even.
+
+\vspace{1em}
+\noindent
+$$\displaystyle{ + \left ( \frac{a}{2^k \cdot n} \right ) = + \left ( \frac{a}{n} \right ) \cdot \left ( \frac{a}{2} \right )^k, +}$$
+where
+$$\displaystyle{ + \left ( \frac{a}{2} \right ) = + \left \lbrace \begin{array}{rl} + 1 & \text{if}~ a \equiv 1, 7 ~(\text{Mod}~ 8) \\ + -1 & \text{if}~ a \equiv 3, 5 ~(\text{Mod}~ 8) \\ + 0 & \text{otherwise} + \end{array} \right . +}$$
+
+\vspace{1em}
+\noindent
+$$\displaystyle{ + \left ( \frac{-a}{n} \right ) = + \left ( \frac{a}{n} \right ) \cdot \left ( \frac{a}{-1} \right ), +}$$
+where
+$$\displaystyle{ + \left ( \frac{a}{-1} \right ) = + \left \lbrace \begin{array}{rl} + 1 & \text{if}~ a \ge 0 \\ + -1 & \text{if}~ a < 0 + \end{array} \right . +}$$
+\vspace{1em}
+
+\noindent
+However, for $n = 0$, the symbol is defined as
+
+\vspace{1em}
+\noindent
+$$\displaystyle{ + \left ( \frac{a}{0} \right ) = + \left \lbrace \begin{array}{rl} + 1 & \text{if}~ a = \pm 1 \\ + 0 & \text{otherwise.} + \end{array} \right . +}$$

\subsection{Power residue symbol}
Received on Mon Jul 25 2016 - 16:57:13 CEST

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