Proof of infinity of twin primes?

[Kassander]

First of all I apologise for the bad English, I am Austrian so my mother tongue is German and I used Google Translate for most of the translations. I also have not mathematical education; this is why this document don´t fulfil the standards for a scientific publication. But I will do my best to present my Theory and give the readers an appropriate understanding of it.

In this document I will describe my Theory of the distribution of prime numbers. This includes a attempt to proof the infinity of twin primes as well as an explanation of the emergence of prime gap´s and prime k-tuple´s. 

 

Known Facts:

"A prime number is a number that is only divisible by itself and one." This is the most common definition of a prime number. But it could also be said; "Prime numbers are the numbers that arise in the gaps of their own multiples". The latter was already taken up by the ancient Greek mathematician Eratosthenes, who used it for his famous Sieve of Eratosthenes. Eratosthenes' sieve is used to find larger and larger prime numbers. This is done by crossing out the multiples of the known prime numbers, which means that the next number that is not crossed out must be a prime number.

It has been known since ancient times that there are infinitely many prime numbers. Euclid proved this with his Euclid theorem. Here is the translated sentence taken from the German Wikipedia: https://de.wikipedia.org/wiki/Satz_des_Euklid

“There are more primes than any given number of primes.

Let the given prime numbers be a, b, c. I claim that there are more prime numbers than a, b, c. Form the smallest number measured from a, b, c ( = Least common multiple ). Let it be DE and add the unit DF to DE. Either EF is a prime number or it isn't. First, let it be a prime number. Then one has found more prime numbers than a, b, c, namely a, b, c, EF. Second, let EF not be a prime number. Then it must be measured from some prime number [VII, 31]; let it be measured by the prime number g. I claim that g does not coincide with any of the numbers a, b, c. If possible, do so, namely, a, b, c now also measure DE; g would then also have to measure DE. But it also measures EF. So g would also have to measure the remainder, the unit DF, while it is a number; this would be nonsense. So g does not coincide with any of the numbers a, b, c; and it is prime by assumption. So you have found more prime numbers than the number a, b, c presented, namely a, b, c, g"

In English, the “least common multiple” of prime numbers is known as "primorial" (https://en.wikipedia.org/wiki/Primorial).

So Euclid recourse to the multiples of the already known prime numbers, in order to identify a point, at which a new prime number can arise. At this very special point, the primorial of all previously known prime numbers, the multiples of the previously known prime numbers are bundled, which means that new prime numbers can arise at the points primorial +1 and primorial -1. The numbers around the primorial are also known as "primorial primes" if they actually are prime Numbers. They are also called "Euclidean numbers" no matter if the number is actually a prime number or not. A Euclidean number of the first kind is a number as described in Euclid's proof just presented, which forms at the point primorial+1. A Euclidean number of the second kind uses the same proof by Euclid, but was first described by the German mathematician Ernst Eduard Kummer. This is why it is also called "Kummer-number"; it occurs at the point primorial-1. Here is the link to the Wikipedia article (https://en.wikipedia.org/wiki/Euclid_number).

The smallest primorial are; 6, 30, 210, 2310, 30030,…

 

Visualization:

In order to facilitate the complex explanation and to avoid misunderstandings caused by translation errors, I use a graphic visualization to support my explanation. The graphic visualization is a modified variant of the Sieve of Eratosthenes. In this, the prime numbers and their multiples are not only crossed out, but also in different colours and with different line directions so that they remain distinguishable.

In addition, the numbers are not actually written in this graphic visualization, as this would only reduce the clarity. A checked background is used; each square represents a number that can be recognized by its coordinates. The top horizontal column contains the digits from 0 to 9. The left vertical column contains the numbers from 0 to 57. So the digits in the top horizontal column number the vertical columns and the numbers in the left vertical column number the horizontal columns. These are the coordinates that give the empty squares their number. In order to know which number the empty squares stand for, the coordinates of the vertical columns must be written after the coordinates of the horizontal column. For example:

(15|3)=153

(0|0)=0

(0|1)=1

(1|0)=10

(10|0)=100

(57|9)=579

With this graphic visualization, all numbers from 0 to 579 can be assigned to a square. Of course, this can be extended as desired. But for my explanation this range of numbers is sufficient. You will then see six stages of the graphic. First it is empty and then the multiples of the prime numbers from 2 to 11 are gradually drawn in. Here is the graphic visualization:

 

 

It is noticeable that I not only crossed out the prime numbers and their multiples in the graphic, but also always the number "zero". If the intervals of the multiples of the prime numbers were not only crossed out with ever larger numbers but also crossed out in the other direction on the x-Axis, the interval of each number would end in zero. This is logical since subtracting any number from itself must always equal zero. But this is also very important in this graphic because the geometric properties, which I will describe in more detail later, also make it clear that once the "multiples" of the prime numbers are bundled, this must also be the case later. After all, the intervals of the multiples are regular. When these are bundled again has already been described. It is always the case in place of a primorial.

The second thing to notice is that the number "one" will always remain uncrossed. Normally it would mean that it would have to be a prime number. But the number "one" is in many respects an exception in the current teaching of mathematics when prime numbers are discussed. Here to, it is a conscious decision to simply leave the number one blank. In summary, it can be said that the numbers zero and one are special in this graphic and many general definitions and descriptions do not apply to these two numbers.

In the further course I will try to present all assumptions and proofs both in a formal way and in a geometric way with the help of these graphics. Since, as I said, I have no mathematical education, I almost assume that my attempt at a formal representation will be incorrect. Nevertheless, I consider it important to at least try to offer a formal and logical explanation, since a purely geometrical explanation would hardly be accepted.

 

Properties of the graphic:

Due to the fact that the multiples of the prime numbers were crossed out in different colours and with different lines, it is possible to identify the prime factors of the larger numbers. Of course only if the prime factors of the number has already been crossed out. For example, the prime factors of 26 are 2 and 13, but in the graph it appears that only 2 is a prime factor. In addition, it is not recognizable how often a prime factor is observed. So the prime factors of 16 are four times 2, but this is not evident in the graphic.

I have chosen the usual decimal notation and visualisation for this graphic. In this, the prime factors of the number 10 can be recognized by vertical lines, i.e. the prime numbers "2" and "5". This applies to any number notation. If I were to choose a binary representation, the prime factor would be "2" and the number 2 would be crossed out vertically. The top horizontal column in this case would consist of 2 vertical columns, each of them would stand for either all even or all odd numbers. 30 is the primorial of 2*3*5. Therefore, the prime factors 2, 3 and 5 would be crossed out vertically if I chose a notation with 30 vertical columns. The top horizontal column in this case would consist of 30 digits to number the vertical columns. This will later become important for understanding Dirichlet's theorem.

 

Representation as a sine function:

The first thing that you will notice is that each primorial is a constantly repeating interval. This is logical since the multiples of the prime numbers are also regular intervals. Since these multiples are all bundled into zero, they must also be bundled again later. This happened in the primorial. There is also an already known way of representing this as a sine wave.

                                      

On the left side you see the notation of the sine formula with pi and on the right side the notation with 180°. The intervals of the prime numbers 2 and 3 are shown. The y-zeros of the respective function represent the multiples of their prime number. If, as can be seen in the following Geogebra representation, several prime numbers are represented, this function can be used in the same way as my graphic. But I think my graphics are clearer:

It becomes clearer when the sine curves are combined into one by multiplication (Multiplication works because multiplying one function term by another must always result in "0" at the zeros, since any number multiplied by zero must always return zero). Then it is no longer possible to recognize the prime factors of composite numbers, but the remaining prime number potentials become more visible, they are the remaining integer numbers on the X-Axis that are not intersected or touched by the sine curve. Here is a representation of the remaining prime number potentials when the prime numbers "2", "3" and "5" are already known:

While I have now shown that my graph can also be presented in a way that is common and accepted in today's mathematics, I have already anticipated the next point, the "prime number potentials". They are the actually important part of the graph and as already described they also appear in the sine function notation of my graph. "Prime potentials" are the numbers that are not crossed out. Thus, my graphic is divided into two areas, just like the sieve of Eratosthenes. The "known prime numbers and there multiples", these are the prime numbers and their multiples that have already been crossed out, on the one hand and the "prime number potentials" on the other hand. The latter include all numbers that are not crossed out. As previously mentioned, the number "1" is special and also never crossed out.

 

Formal specification of the terms:

As far as I know, much value is placed on a formal, logical way of writing theories in today's mathematics. Unfortunately, I understand very little about it, but I will try to represent the "known prime numbers and there multiples" and the "prime number potentials" in a formal and logical manner. Please note that an error on my part in these formulas does not mean that the theory based on them is wrong, but that I probably only made a mistake when trying to create such a formula.

To bring even more order into the system, the "known prime numbers and their multiples" should also be subdivided and a category called "known prime numbers" should be created. Known prime numbers always include all prime numbers up to the largest known prime number. So there must be no prime number in between. P_k are defined as:

P_k={p₁, p₂, …, p_n}

I hope this formal notation is correct. If not, please remember that the P_k represents the primes crossed out, the “known prime numbers”.

Accordingly, I think one should be able to write the "known prime numbers and their multiples" as follows:

If all P_k are multiplied by all natural numbers, the result should include all P_k and their multiples, so the "known prime numbers and their multiples" or P_km. However, I ask you again to ignore any potential errors in the formal spelling and simply rely on the definition I made earlier based on my graphic.

 

Next, it is important to formally define the primordial. In German these are called "Primfakultäten" or translated "prime factorial ". According to German-language Wikipedia, this is its formal definition:

I think it could also be written like this:

P_F = {p₁*p₂…p_n}

If this is not correct, please use the Wikipedia formula. In the further course, however, I will designate the primorial as P_F.

 

Now to the important prime number potentials. These include all numbers that are not multiples of the already known prime numbers P_km, i.e. the numbers that have not yet been crossed out according to the sieve of Eratosthenes. Or which are not yet crossed out in my graphic, i.e. the empty free fields. The prime number potentials are defined as:

So P_p are prime number potentials.

 

Thus the set of natural numbers N is divided into two areas. The known prime numbers and their multiples P_km and the prime number potentials P_p for which it is not yet certain which are prime numbers and which are composite numbers. As soon as a new prime number becomes known, i.e. it becomes part of the set of P_k or P_km, its multiples are no longer prime number potentials P_p and are "switched off". I use the term "switched off" instead of "crossed out" because, unlike Eratosthenes' sieve, nothing is really supposed to be crossed out on a piece of paper.

The prime faculties P_F are thus part of the set of known prime numbers P_km. Then there are special prime number potentials P_p that arise around the prime faculties P_F. I call these Euclidean prime potentials P_E and these are the twin prime potentials for which this document was written. Up to now, as already mentioned, these numbers were either referred to as “primorial primes” if they were identified with certainty as prime numbers, or they were referred to as Euclidean numbers of the 1st or 2nd kind. The smaller of the two was also known as the "Kummer-number". The definition for the Euclidean prime potentials is as follows:

 

Now there is one last new term and category of numbers I want to describe, the "intermediate primes" or P_I. These are part of the set of prime number potentials P_p and are therefore not yet part of the known prime numbers P_k. In addition, the intermediate prime numbers P_I are always smaller than the Euclidean prime number potentials P_E. They are thus the not yet known prime numbers that will form in the prime number potentials P_p and that are smaller than the Euclidean prime number potentials P_E. Only these P_I have the potential to switch off the P_E. They are defined as:

 

 Summary of the relevant figures:

P_k = known prime numbers 

P_km = known prime numbers and their multiples

P_p = prime number potentials

switched off = Transition from the set of P_p to the set of P_km

P_F = primorial or prime factorial

P_E = Euclidean numbers (twin prime potentials)

P_I = intermediate primes

 

 

Graphic representation:

In order to ensure that there are no misunderstandings due to expected errors in the formal representation of the relevant numbers, I draw the respective numbers in an abbreviated form of my graphic below. I use the 30 primorial as basis with the known prime numbers 2, 3 and 5. The green fields are the mentioned numbers. The following graphic illustrations always refer to the 30 primorial. Of course, the formal representation is generally valid and not limited to the 30 primorial.

 

The known prime numbers P_k in this graphic are 2, 3 and 5:

 

The known prime numbers and their multiples P_km in this graphic are 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38 and 39 (“0” is an exception):

 

 

The prime number potentials P_p in this graphic are 7, 11, 13, 17, 19, 23, 29, 31, and 37 (“1” is an exception):

 

The primorial or prime factorial P_F in this graphic is 30 (6 is a primorial but since 5 is part of this graphic the primorial of the graphic has to be 30.  This is beside the exception “0” the only field where all strokes kame together):

 

The Euclidean numbers P_E are 29 and 31 (Since 2, 3 and 5 are the P_k in this graphic, the primorial P_F of this graphic has to be 30, so the shown Euclidian numbers are the numbers around 30)

 

The intermediate primes P_I are 7, 11, 13, 17, 19, and 23 (The P_I have to be always smaller than the P_E and “1” is again an exception. In this case the P_I are all actual prime numbers, this will never be again the case in bigger P_F. But this does not matter):

 

If there is any uncertainty pleas rely on this graphic definition, because this shows exactly what I meant with my explanations. 

 

Primorial pattern in the graphic:

With the clarified terminology, now comes the crucial part of this document. I will now explain what conclusions can be derived from my graph. I will use the 30 primorial for most explanations as this is the clearest. Of course, all explanations are universal for all primorial.

First, it's important to know that each primorial pattern repeats itself indefinitely. This should be obvious by now since the multiples of the prime numbers can be represented as a sine wave and it is also logical since the primorial are made up of the multiples of their prime factors which are infinite. If you now look at the 30 pattern, you will see that it starts again from the beginning at 30 and that the prime factors 2, 3 and 5 are again bundled in 60. Then it will repeat to 90, then to 120, 150, 180,... Here is the graphic of the 30 pattern, the beginning and the end of the pattern is always marked green:

But the first interval of a primorial is always special. That’s why 30 is called primorial P_F and 60 or 90 is not. An Euclidean number P_E appear just at the end of the first interval of an primorial. This is logic because the Euclid theorem can´t be extended to all further repetitions of the interval. But of course it can still happen randomly that prime number twin’s form around the repetitions of a primorial interval. Indeed, all twin primes, except for the first twin prime (3, 5), form at a primorial or at the repetition of a primorial. This becomes clear when looking at the pattern of the smallest, the 6 primorial:

I have only marked the beginning and end of the 6 primorial intervals up to 48, but I think it is already clear here that prime number potentials are only present around the repetitions of the 6 primorial interval. This also shows why these prime number twins form at all. The reason is that there are potentials for prime numbers only around the repetitions of the 6 primorial interval when, according to Eratosthenes sieve, the multiples of 2 and 3 are crossed out. So there is no great mystery behind the formation of twin primes, according to the sieve of Eratosthenes there are simply no other places where primes can form.

 Now that it is clear why twin primes form, it is still important to understand why the other prime k-tuple forms. Before I get into that, it's important to understand one more characteristic of the primorial interval pattern. Primorial interval patterns are always mirrored towards the center. That´s easy to see in the 6 primorial interval (the center is marked by the green line):

But this can be seen in all other primorial patterns:

It is logic that the primorial interval patterns are always mirrored towards the center because when the prime factors are bundled at the beginning and are bundled at the end then the distances on both sides must be the same. To be more precise, 2 is 2 digits away from zero. And as the interval continues past zero, 2 is bundled in zero (as can be seen with the sine waves). This means that        30-2=28 must also be a multiple of two, since 2 must be in 30, since 30 it is the primorial of 2, 3 and 5. The same applies to all other prime factors of a primorial. 3 is 3 digits away from zero and must therefore also be 3 digits away from 30 (this applies of course both upwards and downwards, 3 is a prime factor of 27 and 33). Of course, it also applies to 5, which is a prime factor of 25 and 35. And of course the intervals then continue straight away, 4=2*2. This means that 30-4=26 and 30+4=34 must be multiples of 2. As simple and trivial as this may sound for such small numbers, the statement that every primorial is a reflection of zero has some relevance. One could even see this as an alternative explanation for the formation of prime number potentials P_p  around the primorial. The Euclidean numbers P_E are, in a way, the reflection of 1. As 1 is one digit away from zero, the Euclidean numbers occur at +1 and -1 from the primorial. Since 1 is not prime by definition, this can be used as an alternative proof for the infinity of prime potentials P_p in the Euclidean numbers P_E. However, this is not the promised proof that the twin prime potentials surrounding the primorial are indeed infinite, but a first step.

Here it also becomes clear why prime gaps form and why they keep getting bigger. Around each primorial and its later interval repeats there are prime gaps at least as large as the prime factors that make up the primorial. Below I have some primorial gaps marked green to show what I mean. The primorial itself is marked blue and the Euclid twin potential is marked orange:

In the 6 primorial pattern the primorial gaps are overlap each other:

 

 

In the 30 primorial pattern:

 

We know now that every primorial pattern repeat itself infinitely, that the first primorial pattern is special, that this patterns are mirrored in the middle and that this is the reason why twin primes and also prime gap´s form.

Now comes the next important part that will explain why the other prime k-tuple form. We already know that in the middle of a primorial pattern the rhythm of the multiples turns around. But there is another special characteristic at this point. In the half of a primorial, all prime factors of the primorial are bundled as well as in the primorial itself, except 2 of course. This became obvious when we remember what a primorial is. It is a multiplication of all primes to one biggest prime. Therefore 2 as the smallest prime must be always part of a primorial. A division resolve a multiplication. So if I divide a primorial with 2, the result must show a number with the same prime factors but 2. Here is an example with the 30 primorial:

2*3*5=30            | /2

3*5=15

 

What is easy to understand tells us a very imported thing, when all other prime Factors of the primorial are bundled in the half of an primorial and 2 is not, then the multiples of two has to be at the place +1 and -1 of the half of the primorial. Therefore at the palace +2 and -2 as well as at the places +4 and -4 could not be primes that form the primorial. We remember that the primorial is a reflection of zero, you can always imagen a primorial or a fracture of it as a place where the interval of the bundled multiples start from zero. In this case with the slight modification, that 2 is shifted by one position. Because 2 is the only prime factor of 2 and 4, at the places +2, -2, +4, -4 can be no other prime number that is small enough to switch of this prim number potentials P_p. A intermediate prime P_I of course could switch of this prime number potentials P_p. So this fractional Euclidean numbers (or “Kunze numbers” ) forms infinitely often as well as the original Euclidean numbers, but also as well as the original Euclidean numbers an intermediate prime P_I could switch them off. I have marked them in this graphic of the 30 primorial pattern green, this makes the interval better visible:

 

This is the middle of the 210 primorial pattern that includes 7, even at the places +8 and -8 (8=2*2*2) no prime factor of the primorial pattern is small enough that it can switch the prime number potentials P_p off:

By the way the middle of a primorial is always also the middle of multiples of all smaller primorials but this does not matter.

Sine I used the term “fractional Euclidean numbers” you probably guessed that this phenomenon not only occur at the middle of an primorial but also on other fractions of it. In fact you can divide every primorial with all its prim factors, except the biggest one. This is obvious because if you would divide it with its biggest prime factor you would just reach the point of the next smaller primorial. For example:

2*3*5*7=210    | /7

2*3*5=30

So for example 210 could be divided by 2, 3 and 5 in order to reach primorial fraction. Not only the first fraction bundles all other prime factors of the primorial. It is difficult to describe, I think the best thing is to imagine the x-Axis as a physical object or imagine a long lineal. If you try to break a 30cm lineal in 3 equal big pieces then you have to break it at 10cm and at 20cm. If you try to break a 210cm lineal in 5 equal big pieces then you have to break it at 42cm, 84cm, 126cm, and 168cm. You can always multiply the primorial with a fraction, the numerator of the fraction has to be a natural number from 1 to a number that is 1 less than the number in the denominator. With this calculation you always reach a primorial fraction, a point where all prime factors of the primorial are bundled except the number of the denominator you used for your calculation. These are all fractions and the fractional Euclidean numbers that could be found within the 210 primorial pattern (the fractions are marked violet and the fractional Euclidean numbers are marked green):

210*(1/2) = 105

210*(1/3) = 70                  210*(2/3) = 140

210*(1/5) = 42                  210*(2/5) = 84                  210*(3/5) = 126                210*(4/5) = 168

You can see that in the distance of the denominator of the fraction always a potential of a prime number occur. In the first graphic as mentioned in the chapter before, occur even in the multiples of the denominator prime number potentials. In the graphics 2 and 3, that showed the fractions of 3 and 5, could also be seen that at +1 and -1 is also a strong tendency that prime number potentials arise. This has the same reason as in the original Euclidian numbers; most of the prime factors of the primorial are bundled in the fraction and only if randomly the multiple of the one not bounded prime factor hits the numbers +1 and -1 this prime potential is switched off. You also see the symmetry, in the second graphic 70 and it´s fractional Euclidean numbers are symmetric to 140 and it´s fractional Euclidean numbers. In the third graphic are 42 and 168, as well as 84 and 126 symmetric to each other. This should be logic by now because the primorial patterns are always mirrored by the middle.

So let´s summarise the general conclusions that could be made about the primorial interval Patterns:

1.       Each primorial interval pattern is repeated indefinitely.

2.       Each primorial interval pattern is mirrored in the middle.

3.       The end of a primorial interval is a reflection of the beginning of the interval; this is how the Euclidean numbers and the primorial prime gaps arise.

4.       Euclidean numbers also appear in the fractions of the primorial when divided by their prime factors, these I call fractional Euclidean numbers.

5.       Euclidean numbers or fractional Euclidean numbers appear only in the first primorial interval.

6.       Euclidean numbers or fractional Euclidean numbers are prime number potentials P_p but don´t have to be necessarily primes; they can be switched off by intermediate prime P_I.

7.       Prime potentials P_p may form around the Euclidean fractions at the positions +1 and -1 (since 1 is not a prime number) or plus-minus the denominator ore the multiples of the denominator of the fraction.

There is another conclusion that could be made, I don´t think it´s important but it´s beautiful. I said before that the middle of a primorial is always also the middle of multiples of all smaller primorials. But this goes for all fractions of the primorial. For example at places 1/3 and 2/3 of the 210 primorial pattern so 70 and 140, are also multiples of the places 1/3 and 2/3 of the 30 primorial pattern so 10 and 20. As I said this does not matter but it shows how well those primorial patterns fits together. The way it keeps reflecting and turning into smaller and smaller fractions reminds me of a kaleidoscope. I just wanted to mention that at this point.

It is also interesting that the smallest fraction of a primorial is, from the next smaller primorial, always at the distance of the even smaller primorial times the distance of the new prime number away. For example; 42 is the smallest fraction of the 210 Primorial which includes the prime number 7. 30 is the next smaller primorial, this means that 42 has to be 2 times 6 away from 30 since 6 is the next smaller primorial to 30. It seems to be possible to calculate the next bigger prime number with this trick, but that is not true, you always have to know the distance to the next prime number to calculate it and if you know the distance you could of course simply use an addition to calculate it. But here are a few calculations just to show what I mean:

30+(2*6)=42

(42/2)/3=7

2*3*5*7=210

 

210+(4*30)=330

((330/2)/3)/5=11

2*3*5*7*11=2310

 

2310+(2*210)=2730

(((2730/2)/3)/5)/7=13

2*3*5*7*11*13=30030

 

30 030+(4*2310)=39270

39270/2310=17

2*3*5*7*11*13*17=510510

That could be written simply this way:

 

(30+(2*6))/6=7

(210+(4*30))/30=11

(2310+(2*210))/210=13

 

If you now reducing the fraction with the smallest of the 3 primorial you get an addition:

(30+(2*6))/6=7                | /6

5+2*1=7

5+2=7

 

From a mathematical point of view this is nothing special, but it is interesting that this is visually recognizable in my graphics. If you count the number of the smallest primorial between the next larger primorial and the smallest fraction of the even larger primorial, you get the number of the distance between the largest prime number of the largest primorial and the distance between the largest prime number of the second largest primorial. In the following graphic the two multiples of the 6 primorial between 30 and 42 are market green:

 

The proof of the infinity of twin primes and other prime k-tuple:

This is the most important part of this document and also the reason why it was originally written. Before I get to the actual proof, I have to explain in more detail what the intermediate prime numbers P_I are and I have to explain in more detail why Euclidean numbers only form in the first primorial interval.

Intermediate prime numbers P_I are the reason for the supposed randomness of the appearance of prime numbers. If, according to the sieve of Eratosthenes, the multiples of known prime numbers P_km are crossed out, only very regular prime number potentials P_p remain. But these prime number potentials P_p are just potentials and not definitely prime numbers. The reason is that within these prime number potentials P_p new prime numbers are created whose multiples form a new pattern. So for example if only 2, 3 and 5 are among my known prime numbers P_k, the pattern of the prime number 7 is still unknown to me or in other words 7 is an intermediate prime numbers P_I at this point. But I know that the pattern of the prime number 7 could not form a stable rhythm that switches off the Euclidian numbers or Euclidian fractal numbers infinitely, because the pattern of 7 will become once also part of a bigger primorial and will then be bundled in a primorial and in the fractional of a primorial.

This is one reason why the first interval of an Primorial is special, 7 and other intermediate prime numbers P_I could not form a stable rhythm within the first primorial interval but they can form a stable rhythm in the repetitions of an primorial interval because this repetitions are part of an bigger primorial interval that contains 7. For example is 7 a prime factor of 42 and so it will be in every repetitions of this in the next bigger primorial 210:

42                                           |multiple of 7

210+42 = 152                     |multiple of 7

2*210+42 = 462                |multiple of 7

3*210+42 = 672                |multiple of 7

4*210+42 = 882                |multiple of 7

 

The important thing here is to note that the prime number potentials P_p in the Euclidean numbers P_E and the Euclidean fractional numbers I have described are never permanently switched off by a prime number. Once a prime number switched off these prime potentials P_p, it can no longer do so at the latest when it becomes part of the known prime numbers and thus part of the known prime numbers and there multiples P_km. With this it can be stated:

No prime number can permanently eliminate the prime potentials P_p in Euclidean numbers P_E or Euclidean fractions.

Now there is a second important finding. There are infinitely many Euclidean numbers P_E and Euclidean fractional numbers. This is obvious because there are infinitely many prime numbers and therefore infinitely many primorial. Nevertheless, it can be stated:

There are infinitely many Euclidean numbers P_E and Euclidean fractional numbers.

With these two statements it should now be obvious that there are infinitely many twin primes in the prime potentials of the Euclidean numbers P_E and that some other prime k-tuples form in the Euclidean fractional numbers. It must be noted that the probability that twin prime numbers form in the Euclidean numbers P_E and Euclidean fractional numbers decreases rapidly, because the number of intermediate prime numbers P_I increases rapidly. But the probability can never be 0, since a 0 probability requires a stable pattern that switches off the prime number potentials in the Euclidean numbers P_E and Euclidean fractional numbers permanently.

[Kassander]

Based on this theory, I can even create a function that calculates all prime numbers in a certain number space exactly.

By reconstructing the sieve of Eratosthenes as a function, I can calculate all prime numbers that can also be calculated with the sieve of Eratosthenes.

To repeat, for example, if I crossed out all the prime numbers and their multiples up to the prime number seven, then I already know all the prime numbers in the number range between 7 and 7². This applies to all prime numbers in the sieve of Eratosthenes, if all prime numbers and their multiples up to a prime number P have been crossed out, then all prime numbers between P and P² are known with certainty. The reason for this is that every composite number in this number space can be obtained by multiplying it by a prime number smaller or equal P.

I have shown before that it is possible to represent a prime number and its multiples as a sine curve. This method is already known in mathematics. To repeat, with the function y = Sin((180°/P)*x) all prime numbers and their multiples can be represented as zeros of the function if the corresponding prime number is used for P. And as already mentioned, by multiplying this function term with another function term in which a different prime number is used, a combined function can arise in which the zeros of both functions are combined. Here is the example again:

In this example, the prime numbers 2, 3, and 5 are represented as zeros, as are their multiples. The Function is:

Y=Sin((180°/2)*x)*Sin((180°/3)*x)*Sin((180°/5)*x)

It is obvious that any number of such function terms can be multiplied. The zeros are always preserved because every multiplication by zero must always result in zero. With that alone, the promise of reconstructing the sieve of Eratosthenes as a function is already fulfilled, since in this example all prim numbers between 5 and 5² can be reliably identified. They are all natural numbers that are not Y=0.

But there is a Way to improve this. Strictly speaking, prime numbers do not have to be inserted into the function at “P”. Composite numbers can also be used. If these composite numbers are not larger than P² of the largest prime number contained, then the multiples of this composite number are anyway only in the multiples of the prime numbers that make up the composite number. In other words, at position "P" of the function, all natural numbers from 2 can be substituted to any number "n". All natural numbers from 2 to n only have to be used. Then all prime numbers from n to n² can be recognized; they are the natural numbers in this number space that are not y=0. Here is an example:

Y=Sin((180°/2)*x)*Sin((180°/3)*x)* Sin((180°/4)*x)*Sin((180°/5)*x)* Sin((180°/6)*x)* Sin((180°/7)*x)

As you can see, the composite numbers 4 and 6 have no influence on the zeros of the sine wave in this example. And that explains how all prime numbers in a number space can be calculated with a simple function.

[Kassander]

Since ChatGPT has made huge progress here is an new version of the formal approach to prove the infinite existence of twin primes!