The previous weeks I thought if we cannot find any patterns in
prime-numbers because we do not look at the whole set. Maybe there is a
bigger set which also contains primes-numbers. If we can define such a
set and can filter the normal primes we maybe can define a function
which maps all primes.

First we define a new set which is a subset of the complex-numbers
named Demi Complex Numbers denoted with
$D$.

$\begin{equation}
D = \{ x + iy \mid \{x, y\} \subset Z \}
\end{equation}$

It is now possible to define a new prime like set named the Demi
Complex Primes denoted with
$W$.

$\begin{equation}
W = \{ w \mid w \neq p \cdot q, \{ p, q\} \subset D \setminus \{ 1, -1, i, -i \}\}
\end{equation}$

This set looks like the following:

We can see that the set is symmetric on the zero-axis and on the
diagonal. We now define the function
$f(w) = Re(w) - Im(w) + iIm(w)$
which is basically aligning the diagonals to the verticals.

It is now evident that the diagonals repeat themselves after the line
defined by the function
$r(x) = x + 2$.

Given two sets

$\begin{equation}
P_x := \{ x + iy \mid x > y > 0 \land x + iy \in W \}
\end{equation}$

$\begin{equation}
H_x := \{ x + iy \mid x > y > 0 \}
\end{equation}$

The following needs to be proven:

$\begin{equation}
P_x = H_x \iff x \in W \iff x \in P
\end{equation}$

Further Ideas

The next step is to create a function
$h(x) = P_x$
which predicts either the
$P_x$
or its pattern in a other way.