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Benedict Irwin edited Rules.tex
almost 9 years ago
Commit id: ab991941753b55471dfb9d913b199b8ccd1aa633
deletions | additions
diff --git a/Rules.tex b/Rules.tex
index 74fc6b6..516cfb4 100644
--- a/Rules.tex
+++ b/Rules.tex
...
Leaving Rules
\begin{equation}
\begin{array}
\; & [(a+3b) & mod \; 13 =0 & \wedge & n - 1 \; mod \; 6 = 0] \\ % 1 & 7
\vee & [(b-a-1) & mod \; 13 =0 & \wedge & n - 2 \; mod \; 6 = 0] \\ % 2 & 8
\vee & [(3a+b) & mod \; 13 =0 & \wedge & n - 3 \; mod \; 6 = 0] \\ % 3 & 9
\vee & [(4a+b+1)& mod \; 13 =0 & \wedge & n - 4 \; mod \; 6 = 0] \\ % 4 & 10
...
\begin{bmatrix}
&8 & &5 & & &3 &7,10& & \\
1,2 & &3 & & & & & & & \\
& &1,5&2 &4
& &6 & & & & \\
6 & & & &1 & &2 & & & \\
&5 & & & & &1 & & &2,4 \\
& & &4 & & &
& &6 &1 &5 \\
&
& &6 & & & & & & & \\
&2 & & & & & & & & \\
& &4 & &2 & & & &
& &6 \\
& & &
&3 & &3,6& & &2 & & \\
\end{bmatrix}
\end{equation}
Leaving
Rules Rules, if:
\begin{equation}
\begin{array}
\; & [(2a-b-2) & mod \; 17 =0 & \wedge & n - 1 \; mod \; 16 = 0] \\ % 1
...
\vee & [(4a+b-6) & mod \; 17 =0 & \wedge & n - 3 \; mod \; 16 = 0] \\ % 3
\vee & [(3b+a+3) & mod \; 17 =0 & \wedge & n - 4 \; mod \; 16 = 0] \\ % 4
\vee & [(2b+a-6) & mod \; 17 =0 & \wedge & n - 5 \; mod \; 16 = 0] \\ % 5
\vee & [(5a+b+2) & mod \; 17 =0 & \wedge & n - 6 \; mod \; 16 = 0] \\ % 6
\end{array} \\
\frac{Z}{13} \frac{Z}{17} \in \mathbb{Z}
\end{equation}
...
& & & & & &8 &6,7& & \\
& & & & & & & & & \\
& & & & & & & & & \\
6 & & & & & & & & & \\
& & & &6 & &10,15 & & & \\
& & & & & & & &6 & \\
7 & & & & & & & & & \\
& & & & & & & & & \\
&6 & & & & &
&10,15 & & & \\
& & & &
& & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & &6 & & & & \\
\end{bmatrix}
\end{equation}
Leaving Rules, if:
\begin{equation}
\begin{array}
\; & [(4a-b+7) & mod \; 19 =0 & \wedge & n - 6 \; mod \; 18 = 0] \\ % 1
\end{array} \\
\frac{Z}{19} \in \mathbb{Z}
\end{equation}
\begin{equation}
\Gamma_{23}=
\begin{bmatrix}
...
\\%9,9
\end{bmatrix}
\end{equation}
Leaving Rules, if:
\begin{equation}
\begin{array}
\; & [(9a-b-9) & mod \; 23 =0 & \wedge & n - 6 \; mod \; 22 = 0] \\ % 6
\end{array} \\
\frac{Z}{23} \in \mathbb{Z}
\end{equation}
\section{31}
Leaving Rules, if:
\begin{equation}
\begin{array}
\; & [(2a-3b+1) & mod \; 31 =0 & \wedge & n - 6 \; mod \; 30 = 0] \\ % 6
\end{array} \\
\frac{Z}{23} \in \mathbb{Z}
\end{equation}
\section{37}
For divisibility by $37$, (exceptionally common divisor) the general $\Gamma$ matrix can be constructed below
...
& & & & & & & & & \\
\end{bmatrix}
\end{equation}
All entries have a $6$. So if $n-6 \; \mathrm{mod} \; 36 = 0$, $Z/37 \in \mathbb{Z}$.
\begin{equation}
\Gamma_{X}=
...