Chapter 2 of Prime Technology

Now we describe the other mathematical effects that cause Goldbach pairs to break up, as the graph shows. This also shows why the striking trail is created with its columns in the graph. So also that the reversal effect below the gp Goldbach pairs value is not possible.

First to the prominent columns in the trail of the graphic, and the first effect that increases the Goldbach pairs upwards.


If we look again at the 30s module, there are 15 even numbers in it.

Each of these 15 even numbers position forms a specific convolution. It repeats like a permutation after every 30s module.


In the following overview we show the MEC 30² as 15 checkerboard squares, we call them ikons. They show the specific convolution pattern in the even numbers. With their symmetrical positions, they describe, at the most minimal level, the active primes and their active products facing each other in the convolution. They play a very important role, which we will describe in the course of the 4st chapter of this work.



Overview: 30s cycle of the 15 even number positions.


Every even number figuratively has a specific checkerboard pattern.

The number of pairing determined during folding is 3, 4, 6, and 8 they are highlighted in yellow.

Thus, 3 is the smallest possible pairing that we have already calculated as the smallest gp base.

Open is the pairing of 4, 6, and 8, which we represent and calculate in the following chart as another gp basis.


Again, the numbers in the graph have exactly in the red line of 4, 6, and 8, no "prime divisor" larger than 2, 3, u. 5

(5 <prime divisor <√N). Based on their icons, they show exactly their "gp base value" with their specific folding patterns and pairings. So over the smallest possible 3-pair formation, with the smallest gp base value.

This not only divides the distinctive columns in the graph to count, but also the density of the trail. Thus, the 3 in the 30s ikon cycle is represented eight times and the 8 only once, which also illustrates the density distribution of the pairs in the graph.


Now for the second effect of the upwardly erupting Goldbach pairs:


In the graph, it can be seen at the top that additional Goldbach pairs developed from the four gp underlying stocks.

These erupting even numbers all have larger prime divisors, bigger 2, 3, u. 5 and smaller root of N

(5 <prime divisor <√ N), as their almost identical even numbers exactly on the red lines in the graph.


In order to better understand the underlying system, we show a table with the 30s cycle of convolutions.

Starting from an example number 2,892,360 which is to be divided exactly by 30, without a prime divisor,

(5 <prime divisor <√N).

 The principle of this second effect can already be seen in the following table alone.

All even numbers that have these 5 <prime divisors <√N have a lot more Goldbach pairs like their base value with their numbers 3, 4, 6, 8 identical ikons.This can be seen by looking at them in the table with the comparable 3, 4, 6, gp base values. Note: Striking is also in the table the smaller the prime divisor the more Goldbach pairs.

The Goldbach pairs of 3, 4, 6 GP Underlyings have a proportionality to each other. This proportionality is also shown in the table, as well as for the trace of the graph.


What happens in the system?

Let's take from the table first our example number, the first even number n (1) = 2,892,360 with 8 pair forming positions and with the gp minimum base value = 26,135 Goldbach pairs. The even number n (1) does not have 5 <prime divisors <√N is therefore not divisible by 7 and already has 26,673 empirically determined Goldbach pairs, thus about 2% more Goldbach pairs than the gp underlying.


Now we take an even number that also has 8 pairs positions but is divisible by 7.

The n (2) = 2,892,330, it is 30 smaller than the number n (1).  It also has approx. Gp = 26,135 Goldbach pairs, but since it is to be divided by 7, the Goldbach pairs increase by about 18% over the gp base value of 26,135 with 5,817 to 31,951 Goldbach pairs. As the fluctuation in the following graph shows.

Why, are the additional Goldbach pairs created?

We explain this by an analogy of a railway train and its wagons to be pulled. We imagine that train 7 (prime divider 7)  an infinite number of wagons pulled. With a size of 7 x MEC 30 = 210 and 8 "non primes axes" each axis is a product of 7. The exact description in chapter 3.


(Note: In the book MEC 30 of 2008 and in the notarized entry the analogy with the railway train and its wagon is described as a Schablone).


At a train length of n(2) / 210, there are exactly 13,773 wagons on the track rails. In a folded route 6,886.5 wagons are side by side with 8 "non primes axes". The additional "non-primes axis pairs" form. The now purely in the closed course of the even number n(2) the prime numbers so "move" that the gp base value could still arise approximately 6,886 additional Goldbach pairs.


Empirically determined, there are 5,817 additional Goldbach pairs. It is essential that from the system more and more Goldbachpaare arise, to the gp underlying value, the supposition of Goldbach support. Also for these, additional Goldbachpaaren an approximate value "base value" is to be calculated.


We illustrate the principle of these pairs with the example numbers n (1), n (2), and n (3).

Proportionality is shown by the number n(3), since it does not possess the 8-pair formation like n (1), n (2), but the 3-pair formation with two "small" prime divisors, 7 and 41.

To reminder:

n(1) = 2,892,360  / 30 to parts without remainder, with 8 pair formations, primes divider 0.

n(2) = 2,892,330  / 30 to parts without remainder, with 8 pair formations, primes divider 7.

n(3) = 2,892,386  / 30 to parts with remainder 2, with 3 pair formations, primes divider 7, 41.




First, consider the number n(2) and find the smallest 5 <prime divisors <√N.

In the number n(2) it is the prime divider 7.


Now, we calculate the "towing wagons" from this primes divider.

                 2,892,330 / 7 x MEC30 = 13,773          


We use again the inverse of Euler In to determine the minimal influence of the calculated "waggons" on the prime number shift.

                 13,773 / ln (13,773) = 1,445


Due to the folding, "the wagons" are positioned symmetrically opposite each other and yield the "prims product axis pairs", so we have to divide the number 1,445 by 2.

                  1,445 / 2 = 722.5


Since each opposing wagon has 8 prims product axis pairs, we have to calculate the number 722.7 times 8. Thus, at least 11,560 prime numbers shift to 5,782 Goldbach pairs, to the gp underlying.

                  722.7 x  8 = 5,782


Thus, the number n(1) without prime divisors has almost no increased gp base value of their Goldbach pairs.


This shows in the number n(2) with the prime divider 7 and an additional 5,819 Goldbach pairs. Since it is a system, we use the proportionality for the calculation of the additional Goldbach pairs in the number n(3).

The copy of n(3) from the table above.

The even number n(3) = 2,892,386 which also has a prime divider 7, has in the table the 3 pair position and 9,801 gp Goldbach pairs as the underlying. Plus from the system, primes divider 7, still 2.491 pairs more, thus 12.292 Goldbachpaare.  Since it is the same system, we can use the number n(2) to approximate the additional goldbach pairs of n(3). Over the positions 3, 4, 6 and 8.


The number n(2) = 8, and 5,817 additional Goldbach pairs from the system.

The number n(3) = 3. 

(5,817 / 8) x 3 = 2,181


The additional Goldbach pairs from 2,181 to the gp base pairs 9,801 = 11,982 Goldbach pairs.       

So the number n(3) 2,892,386 had 11,982 Goldbach pairs after our first calculation.

But still 310 Goldbach pairs are missing, from the empirically calculated actual value 12,292.


In contrast to the number n(2), the number n(3) harbors, as already mentioned, another small prime divisor below its root, the 41. This extends the Goldbach pairs by about 318.

Since it is a proportional system we can get the 2,181 by the additional "shift train" of 41 parts.

2,181 / 41 = 53


At 53 we have the 7th part of the "missing" crowd.

53 x 7 = 371


Minus 1 x 53 pairs, from the "shift train" 7.

371 – 53 = 318


Now the gp base value of                    9,801

The   7th "shift train"                            2,181

The 41th "shift train"                               318

Goldbach pairs calculated 12,300, of empirically determined value 12,292 of the even number 2,892,389.

Now we come back to the core that the primes do not arise, but reduce themselves by their products in the red positions. It follows that the primes position their products systematically to product pairs. Thus also prime numbers remain which result symmetrically in the number strand Goldbachpaare. In all areas, from the gp base values to the "moving system of the waggons", we showed that the Goldbach pairs have to be calculated with the primes product pairs. The two following tables with the 30s cycle clarify the principle again..


To the charts / table:

N is the even number to be checked.

A are the active prime numbers.                      B are their active products.

X are the Goldbach pairs.                                   C are the pairs from the active Products.

Now we can enter the values from the charte/ table into the following formula and get exactly the Goldbach pairs of the even number N and that with all even numbers up to infinity. Since they are calculated from the products of the primes.

Since no primes A arise, but reduce themselves in the predetermined positions, the system A is shown as MEC30 enlarged in its product B. Thus, also the pairs C from B are represented by A as Goldbach pairs X.


Simply put, you want to make 10 pairs of 10 red stones and 10 yellow stones. But the system pretends that the first pair must be made up of 2 red bricks. So one is forced in the further course to form a pair with 2 yellow stones. This default forces the primes below the root of an even number that we represented in the formula for the gp base.

The even number 5.040 shows the associated parity, the 672 primes A with their 627 products B. Thus, the parallel parity shows, of 672 B are 184 pairs C, so also from 627 A are thus 184 pairs X Goldbach pairs.

The even number 4.322 with only 3 active prime positions in the convolution also shows the principle of parity.

The parity shows that with the increase of the prime numbers A of the even number to be examined, the products B increase as well. In the same ratio as A and B, the pairs C and the Goldbach pairs X increase.

Therefore, we are able to generate the gp base value of the Goldbach pairs X by the product C, from the primes below the root. These then become mathematically visible as Goldbach pairs within the total given prime positions of the even number N. The key is the complementary specification of the positions by the MEC30².


Even the Littlewood fluctuation does not matter, it is a product of the system.Essential is that every infinitely large even number N has at least one active prime A (except 2, 3, 5) below its root. The, the quantity of the gp base value of the Goldbachpaare X determines. The active primes below the root are determined by this formula, so the Goldbachsche conjecture is correct.

In Chapter 3 of primes technology, we go into detail to prove 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 complete induction. In other words, what can be exactly calculated with the "small numbers" in this example up to 3.99 million also applies to infinitely large numbers..

The reason is that the MEC 30 without "computational effort" iterative forms the following function graph that provides parameters for the formulas.

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© Achim Martin Frenzel Primzahlentechnologie 2008