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Benedict Irwin edited Introduction.tex
about 10 years ago
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The cellular automation like data map was first discovered when analysing a program investigating bound states in a rectangular 2D box. States for a box of dimension $(x,y)$ which were not bound were highlighted on an $(x,y)$ pixel map and a rich structure akin to a diffraction pattern is witnessed. Visual analysis found a subpattern which gave information about the primes when interpreted in the correct method.
The generator for the prime data map is a simple function. For some
$n $n,m \in \mathbb{Z}$, let \begin{equation}
x_0 = (2n-1) \\
y_0 = 1 \\
dx = (2n-1) \\
dy = 1 \\
x_n x_m \to
x_{n-1} x_{m-1} +dx \\
y_n y_m \to
y_{n-1} +dy y_{m-1} +dy.
\end{equation}
If this is made for all
of the $n$
limited (limited by the maximum number you wish to
check.
If at each $(x,y)$ in check) and the
sequence $(x_m,y_m)$ are plotted as a
position is filled
with value 1, then pixel (value of 1) the
pattern map is created as seen in Figure
1 occurs. 1.
This pattern has an interesting interpretation. If all positive integers $m$ sit in the coordinate $(m,m)$ as the strong, bottom line, then thier divisors are diagonally positioned upwards. Hence if there is no filled position between the coordinate $(m,m)$ and $(2m-1,1)$ then that number $m$ is prime.