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Organization of Natural Integers

Or a Schematic View of Prime Numbers

* By Georges Peyrichou*

*(A short summary is at the end)*

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I will not repeat what has already been said about prime numbers.

What follows has no arithmetic pretension because I do not master the mathematical language and even less its formalism. The only claim is to show, using a common language and simple schemes, that the Prime Numbers are not mysterious but very well organized according to a simple principle inducing an infinite world, unattainable but understandable. What follows is not fixed and is regularly modified, corrected, as and when deductions are made by the observation of the phenomenon. Sharing this approach and seeking any collaboration to go further is also the motivation.

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Colouring

Imagine an endless beach composed of pebbles polished by the surf to the point of making them all almost identical.

Or a Schematic View of Prime Numbers

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I will not repeat what has already been said about prime numbers.

What follows has no arithmetic pretension because I do not master the mathematical language and even less its formalism. The only claim is to show, using a common language and simple schemes, that the Prime Numbers are not mysterious but very well organized according to a simple principle inducing an infinite world, unattainable but understandable. What follows is not fixed and is regularly modified, corrected, as and when deductions are made by the observation of the phenomenon. Sharing this approach and seeking any collaboration to go further is also the motivation.

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Colouring

Imagine an endless beach composed of pebbles polished by the surf to the point of making them all almost identical.

The goal consists in differentiating them all by painting each of a single color from the mixtures that can be obtained from the three colors, cyan, yellow and magenta, the number of possible color shades being infinite.

We start by aligning the pebbles.

Then with the help of magic brushes

The left pebble is empty. «It does not exist» but is necessary.

The first pebble keeps its natural color.

A first brush applies a layer of cyan to a pebble on two from the second pebble.

Then the second brush applies a layer of magenta to one pebble on three.

The third brush applies a layer of yellow to one pebble on four which gives green.

The fourth brush applies a layer of a mixture from the first three colors. etc.

To better visualize the process, the alignment of the pebbles is broken down into parallel layers of each paint application.

Figure 3 reads as follows:

Same idea as for

The first row, horizontal, is the result of the final mixtures composed by the colors of the corresponding columns.

The continuous diagonal is composed of the new successive mixtures obtained on the first row as well as the new necessary mixtures obtained from the colors of the diagonal itself according to a specific rule.

The other diagonals with spaced elements are composed of layers of colors already used, unique and definitive, participating in the mixtures.

The first pebble, diagonal and horizontal row keeps its natural color then come the first layers.

Chronologically we follow the mechanism in

The first row represents the pebbles to be differentiated by color. Begins by the non-pebble.

The following rows represent the color layers successively applied.

In the order of application of the layers:

The no-pebble is the «zero» pebble 00.

The pebble 01 keeps its neutral color.

The pebble 02 takes the color cyan and provisionally all those corresponding to the second row is 2 in 2.

The pebble 03 takes the color magenta and all those corresponding to the third row is 3 in 3.

The pebble 04 already has the temporary cyan color.

The fourth layer is yellow (005).

The pebble 05 receives this second layer which makes it permanently green (006). We now have four colors from which we can get a mix for the fifth layer.

For all the following mixtures, the choice is to systematically use the last final color obtained on the first row to color the one of the bottoms of the next column. Otherwise, when the column is empty, a new mixture has to be created from the colors of the continuous diagonal. As we cannot use the green that would be alone in the column to color the pebble 05, we mix the cyan with magenta (2 + 3) which gives a dark blue directly coloring the pebble 05.

For the sixth pebble can be used the green of pebble 04 obtained on the first horizontal row which could not be used for the mixture of the fifth pebble. Whenever no mixture can be possible in a column it has to be done from the diagonal according to a selected rule, etc.

The important thing to remember is that when applying a new layer and the column of the corresponding peddle has not yet received a layer, a new mixture from the colors already presents on the main diagonal is required because two pebbles would have the same color. This diagonal includes both the final mixtures carried forward as well as the «created». By this mechanism we can differentiate the infinity of pebbles by shades of colors and this, regardless of the starting order of the first three colors (CMJ). The effectiveness of the method would be the same with the colors red, green and yellow, (RGB). The first three colors could also be mixtures and the infinite set of colors obtained would be different but the choice is due to the fact that in painting we use the three «primary» colors as cyan, magenta and yellow. According to the same mechanism of this coloring used to differentiate pebbles we could use three letters or even three signs that we would arrange according to a defined rule.

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Natural Numbers

Usually, natural integers are dispatched on the positive part of the line of real numbers. Here, as we are only interested in them, so we replace this line by a juxtaposition of squares aligned symbolizing them:

Then we resume the decomposition in layers. The boxes are either empty, or they carry the number of empty squares plus 1 that regularly separate them.

To transpose the mechanism to natural integers, we replace the colors with numbers and obtain the following schematic table:

To transpose the mechanism to natural integers, we replace the colors with numbers and obtain the following schematic table:

The reading rule is as follows:

The first row will give the result of addition-multiplication operations on parallel rows that induce multiplication-division operations of columns.

In each row the integers are added to the desired column, for example: In the sixth row: 6 + 6 = 12.

12 results from multiplications between the numbers in symmetric pairs of the selected column:

Column 12: 1, 2, 3, 4, 6, 12

either

1 x 12 = 12

2 x 6 = 12

3. x 4 = 12

If a column has only a number, we take its square.

If a column is empty, we add 1 to the previous number, the latter being a prime number.

Other examples:

A - In row 3, all 3 up to column 297 are added or 3 + 3 +…+ 3 = 297.

Column 297 has the following numbers: 1,3, 9,11, 27, 33, 99 giving the multiplications:

1 x 297=297

3 x 99 = 297

9 x 33 = 297

11 x 27 = 297

B - Row 17 adds all 17 to column 289.

17 + 17 + … + 17 = 289.

Column 289 has only one number, 17. Take its square, 289.

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« Comets »

The first row corresponds to the sequence of natural integers and the set of the following is the decomposition.

The second row corresponds to multiples of 2.

The third row corresponds to multiples of 3.

Etc.

Decomposition allows to visualize forms of comets (colored) starting from each square of the corresponding number of the main diagonal.

A second series of comets has for «head» a product of two consecutive integers inserted between the consecutive squares such as 20; 18; 14; 8.

Square-headed comets are sufficient to understand the mechanism although there are infinite categories of comets that interpenetrate. See excerpt from the “Multiple Divider Map”

The multiple dividers of comets form a continuation of the form:

n²; (n² - 1); (n² - 2²); (n² - 3²) … (n² - m²); (2n - 1)

4 ; 2.

9 ; 8 ; 5.

16 ; 15 ; 12 ; 7.

25 ; 24 ; 21 ; 16 ; 9.

Etc.

When a column is empty of multiple-divisor, it corresponds to a prime number. By setting the chronology of the composition of the color mixtures on the number chart, we observe the genesis of prime numbers which is in fact only induced by the organization of natural integers, a kind of «negative». Seen thus, the prime numbers would seem to be only a consequence of the organization of natural integers.

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A Sieve: The Prime Triangle

Or sieve by comparison.

From: n²; (n² - 1); (n² - 2²); (n² - 3²) … (n² - m²); (2n - 1), we can build a sieve revealing the prime numbers by comparing the odd integers composing the external diagonal with the natural integers of the rest of the table:

In the column on the left, we write the sequence of squares.

In the right outer diagonal is the sequence of odd numbers.

The first row has only the "1".

The second row is formed as follows: the 3 of the second row added to the 1 of the first row gives the 4 of the second row.

The third row is formed in the same way: We add 5 to each of the numbers of the previous row: 5 + 4 = 9, 5 + 3 = 8.

For the fourth we will have: 7 + 9 = 16, 7 + 8 = 15, and 7 + 5 = 12.

From left to right are the decreasing sequences of comets:

n²; (n² - 1); (n² - 2²); (n² - 3²) … (n² - m²); (2n - 1)

This triangle reveals prime numbers as follows: any number on the diagonal of odd natural integers that is not found in the rest of the triangle is a prime number.

Each following square integer is also the result of the addition: n2 + 2n + 1

Diagonals (descending from left to right) are sequences defined by intervals equal to twice the square number from which they are derived. (36 is derived from 6, which gives interval 12).

The descending diagonals from right to left are sequences having for reason the number. (20; 40; 60; 80…)

As all the diagonals are logical sequences, both descending from left to right and from right to left, it is theoretically possible to obtain any portion of this endless triangle and thus to find any prime number «by comparison» without testing its dividers. Nevertheless, the practical effectiveness of this screen seems very low.

Example 1: To look for prime numbers between squares 16 and 25, compare all numbers in the range 16-25 of the triangle with the range 15-25 of the diagonal. Here we include even numbers and 5n because the screen is not based on divisibility but on comparison. Obviously an algorhytme would exclude them.

Example 2: Between squares 36 and 49 we compare all the numbers of the range 36-49 in the triangle with the range 37-49 of the diagonal.

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Example 3: Between squares 121 and 144 we compare all the numbers of the range 121 and 144 inside the triangle with the range 121-144 of the last diagonal on the right. In that example integers as 2n and 5n are not concern so they appear in light blue. (Most unnecessary numbers are removed). This unreadable table is only here to visualize the areas of the sieve useful for comparison.

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A « map » of the organization

Every point corresponds to a multiple-divisor. They are all part of a comet of the form:

n²; (n² - 1); (n² - 2²); (n² - 3²) … (n² - m²); (2n - 1)

Or the form:

n(n+1); n(n+1)-(1x2); n(n+1)-(2x3); n(n+1)-(3x4); … n(n+1)-m(m+1); … 2n

So, we see that the apparent chaos is actually perfectly organized as well as the «empty» columns corresponding to the prime numbers.

This distribution depends on that of all natural integers. It is a kind of negative that has no independence. Prime numbers can be considered a consequence of the organization of natural integers. They are like rays passing through filters of multiples of each divisible integer. Strange filters with holes larger than their surface !

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Synchronization – Desynchronization

Visually, on the map of the organization of natural integers, by a phenomenon of desynchronization and partial resynchronization, the multiple-dividers form kinds of pyramids more or less large and regular, the vertices corresponding to multiples of 6.

The tops of the most remarkable pyramids are multiples of 60.

The size of these pyramids depends on the importance of partial resynchronizations between divisors-multiples. A synchronization is a return to the initial state of decomposition of the half-line of natural integers. More a multiple of 60 has multiple-dividers more the pyramid is schematically remarkable as that of figure 11 corresponding to the number 465585120 having 1152 dividers and especially all those from 2 to 22. This number is the length of what can be considered as a height 22 module that will repeat every 465585120n.

The Modules

After decomposing the half line of the integers into rows we will now assemble them one after the other to obtain modules that will repeat themselves to infinity.

The simplest module is length 2 and height 2:

Next one is a module with length 6 and height 3

The module of length 60 and thickness 6 can be considered as basic one if we refer to the map of the organization showing the regular spacing between the characteristic pyramids:

Module shortcut in this way

The length of these modules increases rapidly. For a height of 13, the length is 360360, shortcut in this way:

The last five grey rows are not part of the module. They are there to show the desynchronization they cause compared to the first thirteen rows entirelysymmetrical.

The left corner is the zero.

The right corner is 360360

The symmetry is in 180180.

Here 180179 and 180181 are prime numbers but do not extrapolate and imagine that all 180180n plus or minus 1 are first; What probability?

Each natural integer being declined to infinity in multiples, a theoretical infinite module exists… with its symmetry.*(Note 8)*

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The Modules and their interest

The modules have a chronological interest and allow to better understand the mechanism of the prime numbers. We have seen that the sieve by comparison does not use division or multiplication but only iterative addition which means that any natural integer is a potential prime number and that a priori we do not know which natural integer is divisible into two equal integers. Any integer can then be considered as a potential prime number. So, by juxtaposition of iterative rows, as with the method of magic brushes we will eliminate progressively potential prime numbers. Stacks of row (height) form a module with length determined by its symmetry. Each module is unique.

Remember that we reason about boxes, columns and rows and that we consider only the set of natural integers, then here discontinuous. Between two consecutive integers (box), there is nothing. Between two non-consecutive integers there are one or more consecutive empty boxes, kinds of empty units. (Here, by definition a unit is note breakable).

We will use again the chronological method of magic brushes. With the colouring, a module leaves coloured boxes and others with no color. These potential primes will be covered or not, only by the rows of the following prime numbers. Now we consider the scope and the range of a prime iteration, which will reach its first column. It colors its corresponding box of the row of results on the first row. This iteration of «n» will fly over already marked columns to reach its second blank column in n2 + 2xn. Then according to the unique size of its module the iteration will continue its eliminations, leaving behind it the part of a module of height n between n and n2 + 2xn where the integers are definitively considered prime. (Confirmed Prime Area). In this area the iteration only covers already blocked columns such as even columns or by those of lower prime number iterations. An even iteration only covers already occupied columns, while a prime iteration eliminates the potential primes.

Each repetition of a prime module eliminates a precise number of prime potentials at the same symmetrical positions relative to the middle of the module.

For the module of 5 we will have the spaces:

5 - 20 - 10 - 20 - 5

*(See length 60 and height 6 module)*

For the module of 7 we will have the spaces:

7-42-28-14-28-14-28 42 - 14 - 42-28-14-28-28-42-7

A module corresponding to a prime number leaves behind it a quantity of validated prime numbers for the reason they will never be covered again because of the longer iteration size of all the next prime numbers.

The size of the modules grows fast and therefore the CPA part too. Larger is the module, the more you have possibility of prime numbers to remain.

At the same time, the gap between consecutive squares increases slowly but steadily unlike the 2xn part. The CPA part and the interval between the two consecutive squares m2 and (m + 1)2 containing one of the prime numbers n are related. The increase in the size of their differences being in favor of the CPA, not eliminating potentially prime integers, makes the prime numbers between two consecutive squares increase constantly, what can be seen by observing the decrease in efficiency of the modules to eliminate the potentially prime integers. This can be verified by analyzing the drop in elimination of potentially prime integers from each new prime module, i.e., sequences unique to each module for example (5 - 20 - 10 - 20 - 5).

The series of modules, in height and length dimensions, does not respond to a simple logic, on the contrary because it is directly related to desynchronization – partial resynchronization. It cannot therefore be useful for an easy understanding of the phenomenon of prime numbers.

The next illustration, unfortunately not readable by its large width can still be understood: The dense and coloured central row corresponds to the result of the elimination of potentials prime number by rows 2, 3, 5, 7, 11, 13. The rows below are CPA n2 and 2xn adjacent. To the same rows above have been added the prime numbers not eliminated by the corresponding prime number. Just above the coloured row are the squares. Be careful, the fact that the prime numbers increase in quantity in the intervals between consecutive squares going so increasing is observed only from the squares of the fifty on a sliding average.

The left corner is the zero.

The right corner is 360360

The symmetry is in 180180.

Here 180179 and 180181 are prime numbers but do not extrapolate and imagine that all 180180n plus or minus 1 are first; What probability?

Each natural integer being declined to infinity in multiples, a theoretical infinite module exists… with its symmetry.

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The Modules and their interest

The modules have a chronological interest and allow to better understand the mechanism of the prime numbers. We have seen that the sieve by comparison does not use division or multiplication but only iterative addition which means that any natural integer is a potential prime number and that a priori we do not know which natural integer is divisible into two equal integers. Any integer can then be considered as a potential prime number. So, by juxtaposition of iterative rows, as with the method of magic brushes we will eliminate progressively potential prime numbers. Stacks of row (height) form a module with length determined by its symmetry. Each module is unique.

Remember that we reason about boxes, columns and rows and that we consider only the set of natural integers, then here discontinuous. Between two consecutive integers (box), there is nothing. Between two non-consecutive integers there are one or more consecutive empty boxes, kinds of empty units. (Here, by definition a unit is note breakable).

We will use again the chronological method of magic brushes. With the colouring, a module leaves coloured boxes and others with no color. These potential primes will be covered or not, only by the rows of the following prime numbers. Now we consider the scope and the range of a prime iteration, which will reach its first column. It colors its corresponding box of the row of results on the first row. This iteration of «n» will fly over already marked columns to reach its second blank column in n2 + 2xn. Then according to the unique size of its module the iteration will continue its eliminations, leaving behind it the part of a module of height n between n and n2 + 2xn where the integers are definitively considered prime. (Confirmed Prime Area). In this area the iteration only covers already blocked columns such as even columns or by those of lower prime number iterations. An even iteration only covers already occupied columns, while a prime iteration eliminates the potential primes.

Each repetition of a prime module eliminates a precise number of prime potentials at the same symmetrical positions relative to the middle of the module.

For the module of 5 we will have the spaces:

5 - 20 - 10 - 20 - 5

For the module of 7 we will have the spaces:

7-42-28-14-28-14-28 42 - 14 - 42-28-14-28-28-42-7

A module corresponding to a prime number leaves behind it a quantity of validated prime numbers for the reason they will never be covered again because of the longer iteration size of all the next prime numbers.

The size of the modules grows fast and therefore the CPA part too. Larger is the module, the more you have possibility of prime numbers to remain.

At the same time, the gap between consecutive squares increases slowly but steadily unlike the 2xn part. The CPA part and the interval between the two consecutive squares m2 and (m + 1)2 containing one of the prime numbers n are related. The increase in the size of their differences being in favor of the CPA, not eliminating potentially prime integers, makes the prime numbers between two consecutive squares increase constantly, what can be seen by observing the decrease in efficiency of the modules to eliminate the potentially prime integers. This can be verified by analyzing the drop in elimination of potentially prime integers from each new prime module, i.e., sequences unique to each module for example (5 - 20 - 10 - 20 - 5).

The series of modules, in height and length dimensions, does not respond to a simple logic, on the contrary because it is directly related to desynchronization – partial resynchronization. It cannot therefore be useful for an easy understanding of the phenomenon of prime numbers.

The next illustration, unfortunately not readable by its large width can still be understood: The dense and coloured central row corresponds to the result of the elimination of potentials prime number by rows 2, 3, 5, 7, 11, 13. The rows below are CPA n2 and 2xn adjacent. To the same rows above have been added the prime numbers not eliminated by the corresponding prime number. Just above the coloured row are the squares. Be careful, the fact that the prime numbers increase in quantity in the intervals between consecutive squares going so increasing is observed only from the squares of the fifty on a sliding average.

Remember that:

That a module is a summary of the previous layers that can be transposed at any level by adapting it to partial synchronizations. These reports have no predictive effect because each next row will form a new module whose length cannot be known without first finding the next prime number thanks to the comparative sieve.

That the way to eliminate the potential prime numbers depends entirely on the desynchronization of the rows hence the unknown in n2 + 2xn.

That only modules whose last row is a prime number eliminate the potentials primes. The elimination distribution is unique for each module, it is symmetrical with respect to the horizontal center of the module and the gaps between the eliminated integers are pairs of the prime number of the row.

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That a module is a summary of the previous layers that can be transposed at any level by adapting it to partial synchronizations. These reports have no predictive effect because each next row will form a new module whose length cannot be known without first finding the next prime number thanks to the comparative sieve.

That the way to eliminate the potential prime numbers depends entirely on the desynchronization of the rows hence the unknown in n2 + 2xn.

That only modules whose last row is a prime number eliminate the potentials primes. The elimination distribution is unique for each module, it is symmetrical with respect to the horizontal center of the module and the gaps between the eliminated integers are pairs of the prime number of the row.

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A choice of reference

The mechanism of prime numbers seems chaotic. The decomposition of the succession of natural integers juxtaposed into an infinity of alignments of its components highlights an organization of the squares n2. The continuation they induce is a «constant» evolving according to a simple and regular rule.

The observation of the extract from the divider map*(Figure 10)* shows the simple shapes of comet varieties and particularly those whose «head» corresponds to a square.

The choice of comets with square heads can therefore serve as reference and allow an attempt to understand the distribution of prime numbers as we advance in the levels of the sizes of integers.

The reference to the difference between two consecutive squares is justified by the fact that this difference evolves regularly. The density of prime numbers also evolves, irregularly but while respecting a contained irregular range. In the first intervals between successive squares, the quantity of prime numbers varies, sometimes decreasing by one distance to the next. But the trend is confirmed with large numbers because the principle is iterative. Remember that if the density of prime numbers between two successive squares decreases, their number increases on average. This density evolution is continuous and there is no break between the squares. At all levels a range is respected. Even the order of occurrence of the gaps sizes, which is irregular, is “framed”. (A gap size is the difference between two successive prime numbers: 2, 4, 6, … 2n.) We are in the presence of an irregular induced phenomenon, which is in fact self-regulated by the regular iteration from which it comes. So, we decide to choose for reference the gap between two consecutive squares while keeping in mind that the phenomenon is slippery.

The mechanism of prime numbers seems chaotic. The decomposition of the succession of natural integers juxtaposed into an infinity of alignments of its components highlights an organization of the squares n2. The continuation they induce is a «constant» evolving according to a simple and regular rule.

The observation of the extract from the divider map

The choice of comets with square heads can therefore serve as reference and allow an attempt to understand the distribution of prime numbers as we advance in the levels of the sizes of integers.

The reference to the difference between two consecutive squares is justified by the fact that this difference evolves regularly. The density of prime numbers also evolves, irregularly but while respecting a contained irregular range. In the first intervals between successive squares, the quantity of prime numbers varies, sometimes decreasing by one distance to the next. But the trend is confirmed with large numbers because the principle is iterative. Remember that if the density of prime numbers between two successive squares decreases, their number increases on average. This density evolution is continuous and there is no break between the squares. At all levels a range is respected. Even the order of occurrence of the gaps sizes, which is irregular, is “framed”. (A gap size is the difference between two successive prime numbers: 2, 4, 6, … 2n.) We are in the presence of an irregular induced phenomenon, which is in fact self-regulated by the regular iteration from which it comes. So, we decide to choose for reference the gap between two consecutive squares while keeping in mind that the phenomenon is slippery.

Prime Numbers Density

With the comparison sieve we observe that the density of prime numbers decreases towards large numbers, and that their quantity between two successive squares increases, on average.

As for the frequency of gaps sizes between prime numbers, from a relatively high level, it still remains within an increasing range. As density decreases, possible types of gaps increase and any size of gap persists. New types of gaps appear constantly. Their order of appearance is not regular but remains bounded within a range corresponding to their level. No size of gap between prime numbers disappears by iterative construction of the «map» of the multiple-divisors.

To observe the distribution of prime numbers the simplest is to do it in the regularity of the range between the consecutive squares.

The most common gap is 6, although decreasing in density as the gap between consecutive squares increases. Then come in decreasing quantity the multiples of 6 as picks between which in lesser number the intermediaries 4, 8 ,10, 14, 16, 20, etc. The gap of 2 remains one of the most frequent, but up to what level? Only small samples levels up to 1015 were tested. The following graph is an example of a sample taken from the 1012 area; (4293 out of 69432 prime numbers between 9746792 and 9746802 consecutive squares)

With the comparison sieve we observe that the density of prime numbers decreases towards large numbers, and that their quantity between two successive squares increases, on average.

As for the frequency of gaps sizes between prime numbers, from a relatively high level, it still remains within an increasing range. As density decreases, possible types of gaps increase and any size of gap persists. New types of gaps appear constantly. Their order of appearance is not regular but remains bounded within a range corresponding to their level. No size of gap between prime numbers disappears by iterative construction of the «map» of the multiple-divisors.

To observe the distribution of prime numbers the simplest is to do it in the regularity of the range between the consecutive squares.

The most common gap is 6, although decreasing in density as the gap between consecutive squares increases. Then come in decreasing quantity the multiples of 6 as picks between which in lesser number the intermediaries 4, 8 ,10, 14, 16, 20, etc. The gap of 2 remains one of the most frequent, but up to what level? Only small samples levels up to 1015 were tested. The following graph is an example of a sample taken from the 1012 area; (4293 out of 69432 prime numbers between 9746792 and 9746802 consecutive squares)

Abscissa are the types of gaps between two successive prime numbers. They range from 2 to 2 such as 2, 4, 6, 8, … 284, … to infinite even.

In ordinate is the quantity for a sample.

The first gap in frequency of occurrence is 2 (twins).

The gap of 4 follows then comes the peak of 6.

Followed the gaps of 8 and 10, smaller and the second peak of 12.

Gap types do not always appear in ascending order.

All peaks are multiples of 6 that go decreasing.

Prime numbers are potentially contiguous to multiples of 6.

Between two peaks of 6n the values of deviations are less frequent.

The number of types of possible deviations between two consecutive squares increases as well as their frequency of occurrence.

No type of gap disappears although their density decreases. The slow evolution of this curve is confirmed until 1015. Refining on continuity and checking beyond to be convinced of it remains to be done.

The choice to observe by pointing to the squares is only practical, the phenomenon of the evolution of the gaps in quantity and type being slippery.

Integers and sinuses

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Footnotes

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