Chapter 3 of Prime Technology

From here we go into detail for the proof that our formulas are infinitely valid.

Thesis: The MEC 30 determines the distribution of primes and their products, thus the number of Goldbach pairs.

For the proof we use the especially induction. What exactly can be calculated with the "small numbers" here in these examples up to 3.99 million, also applies to infinitely large numbers.

The reason is that the MEC 30 forms the function graphic by iteration.

That is, an infinite number simply increases the value of the MEC 30 system.

Thus, the infinite number to be examined determines the size of the function graph. Its internal structure of the graph is determined by the MEC 30, which provides parameters for the formulas through iteration and self-similarity. Thus, the gp base value greater than zero indicates the Goldbach pair, since this is determined by the parameters of the active primes below the root.

It was obvious to study mathematics for their own operating system.

Considering the first prime number 2 and its products up to 30, they will continue from 30 to 60, and so on until infinity.

Considering the second prime number 3 and its products up to 30, they will also continue from 30 to 60, and so on until infinity.

Considering the third prime 5 and its products up to 30, they will continue from 30 to 60, and so on until infinity.

This creates a symmetrical order in positions that could are not to be divided by 2, 3 and 5.

This symmetric order in the first 30s module continues indefinitely. It should be noted that only in the first 30-module, the information and the position are the same number.

The whole is written by mathematics itself, from the products of the first three primes.

The result is that only the numbers 1, 7, 11, 13, 17, 19, 23, 29, are in the red shaded positions, could are not to be divided by 2, 3 or 5.

Referring to Fig. 1, we continue to call them prime positions regardless of whether they are primes or not.

Behind it hides an operating system, the prime numbers 2, 3 and 5 form the framework of the system, it carries the 30s module with the red prime positions. With that we are able to logically ignore 73.33 ...% of all system numbers. Since it belongs as a product of 2, 3, and 5 to the framework of hardware in which the operating system is implied, that we will show here,

1. Calculus: The MEC 30 arises from the behavior of the first 3 primes.

To understand the system, it is of utmost importance that we only refer to the numbers as primes and their prime products that are in the red positions. Because they form the program of the operating system. So, the 2, 3, 5, how to exclude their products and to consider the 1 as a part of the red prime positions.

This is the core of the operating system, the MEC 30

"Mathematical Elementary Cell 30"

The MEC 30 is presented in 2 basic variants, as information and position.

1. as information.

2.   as position.

1. Axiom: The numbers consists of information and position.

2. Axiom: The information and position are equal in the first module of the MEC 30.

If one assumed that no program existed for the operating system of the prime numbers, prime numbers would be in all red positions primes. Since all other positions are 73.33 ...% by 2, 3, 5, allot. As an example, assuming 300,000 / 30 = 10,000 modules.

10,000 modules x 8 red prime positions = 80,000 prime numbers, ie 26.66 ...%, and at the same time 40,000 Goldbach pairs would be possible. "Of course it's not that easy."

But this thought-game forms the most important statement for the prime-number operating system. The primes reduce themselves in the primes positions, no primes are generated, but they are reduced by their products themselves, in the red primes positions.

3. Axiom: The positions for primes are generated symmetrically with the iteration of the Modular 30, the MEC 30.

Although there are 26.66 ...% red positions but not all primes, since their prime-number products also position themselves in these positions. Now comes the kernel of the operating system, only the primes as factors in the red positions, with the factors of the primes from the red positions, can place their products only in the red positions. If it were not so, the primes would be divided as factors themselves by 2, 3, 5, which can not be the case. Thus, the products in the red positions can not be divided by 2, 3, 5, without rest. This continues indefinitely, since the products, the products of the "red" primes come.

2. calculus: Without the prime numbers 2, 3, 5, and their products a graphic is now producing. The MEC 30 determine the prime numbers by iteration and permutation in the graphic.

3. calculus: Entering the MEC 30 through iteration and permutation is the basis for a full induction, proof of Goldbach's conjecture.

With the following function graphics we visualize the operating system of the primes.

The function graphic consists of two parts, the first part, it is the upper gray part in which the MEC 30 is entered as information. The second part, is the lower red part, in which the MEC 30 is to be entered as positions. In the middle runs the number string to be examined, in which the interplay of the two variants of information and position determines the number and positions of the prime numbers. It also shows that the MEC 30 is registered by iteration (regular insertion without computational effort).

In the further course the function graphic will be described step by step. At the end of this chapter and in Chapter 4, the function graph is shown in full with the even number 2.520. The value of the even numbers we represent in the function graph is not random, but corresponds to the size of a finite number that can still be visualized on the A4 sheet.

Overview of the function graph for a finite number.

As already mentioned, the values in the upper and lower part are not "calculated" but entered by iteration and self-similarity. This allows us to enlarge the function graph infinitely with the same rhythm.

Now we show in 3 steps the interplay of positions and information.

Step 1: We describe the function of the lower red part of the graph and show how the positions work.

In order to be able to represent this area of the operating system, the MEC 30² forms a logic gate arrangement, a cell matrix which can also be found in a "CPU". It has the task of displaying the position of the primes and their redundancy-free products in the function graphic. And is to be entered in the lower part of the function graphic.

We now manufacture the first cell matrix with MEC 30². From 1 to 841, in the left square. It still carries the information of the numbers. In the right square we show the products in positions as Modular 30, Example 841/30 rest 1.

The red right square, we call Ikon. It carries the positions of the numbers.

Looking at the next products from the second 30s module, from 31 to 60, we get the information of the products from 31 to 1.711 in the left square.

The right square again shows the identical icon. It again only carries the position of the numbers.

Note: to the analogy with the "train" in Chapter 2, the size of the wagons:

The 7er wagon is 7 x 30 = 210.

Thus, the 37er wagon

37x30 = 210 + 900 = 1,110.

MEC 30² = 900.

The products of the primes move in a permutation (arrangement of objects in a particular order) that the MEC 30² forms as a cell matrix.

This permutation of the red icon is now to be entered by iteration (process of repeated repetition) in the lower part of the function graphic. This makes it possible to iterate all positions of the products to infinity without having to use a computer.

The architecture of the cell matrix MEC 30² is to be entered as a permutation by iteration for an infinite number in the function graph.

Now let's go into the detail of the lower part of the function graph.

The first ikon shows the yellow row of numbers, it can only be divided by 1 and continues indefinitely. This row thus contains all prime positions of the assumed even number of 26.66 ...%. In these prime positions, only the prime numbers A and their products B are positioned.

Since we have built the function graph without 2, 3, 5 and their products, we can exclude 73.33 ...% of the numbers to be examined.

Thus, for a finite number N over this yellow series of numbers, the prime numbers A are to be computed if their products B are known.In the next steps, we want to show how the primary products B registers themselves by way of the MEC 30². We use the primes, reduce them self in the primes positions.

Calculation of prime numbers A, by their primes products B in the primes positions.

Thus, it is important to determine the primes products B exactly and redundancy free.

In the right ikon we show the system how the first redundant systems logically infinitely reduced, they are highlighted in gray in the Ikon. This clears the way to filter out the remaining redundants in the remaining red area in order to determine the exact number of primary products B with the MEC 30.

Greetings to the University of Stanford USA

The special feature is that in this icon we show that the prime number products always start more often in positions 1 and 19 ("the train with its wagons"). This means that the products start more often as numbers with 1 or 9 than with 3 and 7. Similarly, you can find more primes that follow on 3 and 7 than with 1 and 9.

This phenomenon was discovered by the professors of the University of Stanford USA 2016.

Link to the article in the New Scientist. Their publication shows, however, that they did not understand the system behind it. For this alleged discovery of the phenomenon they needed the mainframes of the United States. We discovered the system behind it back in 2008, parallel to the helical structure of the light. Described in the book MEC 30 that I published in 2008 at the Frankfurt Book Fair. I went further in the book. And used the simplest representation of the prime number position with 1, 3, 7, 9 to calculate the electrons of the base pairs, the DNA..

To filter out more redundant, we show how the MEC 30 is used to enter the ikon into the graph with a finite number. To do this, we once again use the analogy from the train that positions its wagons (red) with 8 primes product axles each on the track (yellow) of the primes positions. Thus, each axis, as a product of the primes, represents its position as not prim

As already mentioned, the yellow-marked number series to infinity can only be divided by 1.

It does not make the train, but the track in our analogy (1x MEC 30 = 30). In which each module of 30 corresponds to a rail on which 8 axis positions each can fit.

Since we examine the number 2.520 in the function graph, the "yellow rails of the tracks" are now entered 84 times (iterated) in the functiongraph, shown in the black framed box. Thus, the total track = MEC is 30 x 84 rails = 2,520 so that 627 primes positions "Axis Positions."

On this track with 84 rail units is now under it the 7er train with its waggons in the size of 7 x 30 = 210 and 8 axles each. Note: These axles of all wagons represent the primes product positions B.

See the size of the wagons from the information icon above (MEC30² = 900).

On the route of 2,520 are 12 wagons in the length of 210 with 8 axles each.

Now the 12 wagons are not entered according to their size 210, but according to the number of axles.

To transfer this train, we only need 12 "yellow rails" 8 axle positions for the 84 yellow rail units.

On this route there are other trains, such as the train 11 with its wagons in the size 11 x 30 = 330 and 8 axes each.

On the route of 2,520 are 7.63 wagons in the length of 330 with 8 axles, etc.

Now the 7.63 wagons are re-registered according to the number of axles. There are 62 axes so we need 7.63 "yellow rails" with 8 axis positions.

In the upper functional graphic, the entire 11er train is framed in black with 7.36 wagons and 62 axles and is shown enlarged in the lower part.

It is essential that the wagons of trains in this area are registered only with their 8 axles. So waggons widening with their limited 8 axes in the functional graphics, in the gray redundant area. Thus, the number of "trains" is limited by the root of the numbers to be examined, √N.

As an example of the number 2520 in the function graph = √2520 / 3.75 = 13 moves

There are 14 positions minus the 1 as track, thus 13 "trains". So stand at the route length 2520, 13 trains. Train position 7, 11, 13, 17, 19, 23, 29 | 31, 37, 41, 43, 47, 49 with the corresponding sizes of the "wagons".

Note: The 49er move is part of the 7 move, and thus a redundant move that reduces the calculations to 12 moves, as we'll see shortly.

Thus, we represent the primes products B of a number to be examined, above the root, as "axes of the trains which, so to speak, stand side by side with one another". These "parallel trains" of the red primes products B are not random, but arranged in a systemic manner among each other. This ensures that systematically the remaining redundants are filtered out by iteration, as well as the redundant "trains".

Step 2: Now we describe the function of the upper part of the graph and show how the information works.

Information.

Information. In this upper part of the functiongraph we enter the "wagons of the trains" in their size and not in the number of axles. Example: First train in the wagon size 7 x 30 = 210.

Note: We also use the reduction of 73.33 ...% (by removing the numbers 2, 3, 5, and their products) so that we can reduce the 210 Waggong to 26.66 ...% and represent 56 yellow positions in the functiongraph.

So are the following trains, like the second train of 11 with the wagon size 11 x MEC 30 = 330 large, but is entered by 73.33 ...% smaller by iteration in the upper part of the graph, etc.

Now, the self-similarity with the MEC 30 allows us to carry these "wagons" through iteration into the functional graphics to infinity.

The black-framed MEC 30 (26.66 ...%) is entered seven times larger in the number space of the function graphic. And converts the positions of the MEC 30 in their 7er products. These products mark the positions where the redundants can be found in the lower red area of the graphic. See black arrow

If we now draw one line down from the "axles, the wagons" in the upper picture, marked with the black arrow, it will cross the "track bed 1". There it marks the primes product B in the yellow primes positions. If we extend the line further down into the red area, it marks the other redundants in the "side-by-side trains".

I

In this further enlarged illustration, we again show the principle of the marking of the redundant in the lower "trains". Note: the redundants in the lower "trains" always start below the same "train number" of the upper output train. As in the pictorial representation, in the upper output train 7, the axis is represented as information 49 in position 19. Thus, the redundants start in the lower area not in the train of seven, but always in the train below, in this example, the 11er Train etc.

Since there is no other train over the 7th train in the red area, the 11er train is always the initial train for the first Redundants.

It follows that in the upper part of the function graph, with a finite number like 2520, only 3 moves need to be entered to calculate the last redundancies. Calculation example in 4 steps.

Step 1: Determining the Size of the "Train of the 11th Train".

11 x MEC 30 = 330

Step 2: Number of iterations of the "11er wagon" in the finite number N.

2520/330 = 7.6363 ...

Step 3: Determination of the size of the 11 "train" in the function graphic.

7.6363 x 30 = 229.09

Step 4: Calculation of the possible "trains" below the root of the 11th "train" in the functiongraphic.

√229,09 = 15,13 thus 7, 11, 13 = 3 moves.

Note: A 49 "train" is also a redundant of the 7 "train" in the function graphic (marked with the red arrow in the upper diagram). This too reduces the  iteration ing as numbers increase.

Step 3: This shows the interaction of the information and position in the function graphic with a finite number.

Here is the complete functiongraphic.

The iteration and self-similarity of the MEC 30 shows how the number of primes can be calculated. It shows also in a short clearer form, the yellow positions of the first primes. Those who reduced themselves in the primes positions by their products themselves.

Calculation of prime numbers A by their primes products B in the prime positions.

Below the 672 primes positions of the first "yellow" row, the "moves" are positioned with the red "axis positions". These axis positions of the primes products are shown compressed as "trains". Folded out, they are distributed over the entire 672 prime positions, the number to be examined, 2520, thus 307 primes products B at 672 prime positions.

This confirms our core statement that the primes reduce themselves in the primes positions.

307 primes products B in the 672 prime positions reduce the possible "672 prime numbers" to 365 prime numbers A.

In Chapter 4, we introduce the 15 complementary-structured ikons into the functional graphic by iteration. Thus we show the complete induction also for the Goldbach pairs.

Example graphic:

Link to the Chapter 4 of Prime Technology

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© Achim Martin Frenzel Primzahlentechnologie 2008