Demi Complex Primes

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.