Prime Numbers Distribution Line

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2 ffiffiffiffi N p : (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2 ffiffiffiffi N p integers, that is, Р3–Р1 < 2 ffiffiffiffiffi N p integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than

Alternance ≤Р 1 is the section of P.I. in the form of array of primes -NOT residues of mod(Р 1 #), (for the 1 least common factor > 1 from every NOT residue).

Purpose and role of the overall length of the of alternance (array) of the all first primes ≤Рn
It is quite obvious that φ(Р n #) of the least residues of mod(Р n #) type = С and (Р n #-С), of every recurrent fractal = Р n #, gradate P.I. as φ(Р n #) "eliminated" sections of P.I. with different lengths of the type: С..3рр3.С.3рр3.С 3рр3 С., with ..3рр3.. "eliminated" sections of P.I. represented as array of "eliminated" NOT residues of mod(Р n #), or un the form of alternance (array) of the first primes ≤ Р n , (according to the 1st least prime factor >1 from every NOT residue of mod(Р n #)), hereinafter the alternance ≤Р n . С -residue of mod(Р n #) (according to the 1st least >Р n from every residue of mod(Р n #)), location from 1 to Р n # is line symmetrical relating to number = Р n #/2. And further, repeated without rearrangement of their position with the period = Р n #. Then, after we define the overall -maximal length of alternance that we can form using the fist primes р ≤ Р n , (NOT residues of mod (Р n #)), type С 1 … 3рр3рр3рр3 … С 2 that is maximal amount of consequent odd numbers = maximal length of alternance -р ≤ Р n (one least NOT residue of mod (Р n #) >1 from the number), we can evaluate the distance between every two consequent residues of mod (Р n #) that is between two primes <(Р n+1 ) 2 , according to formula: (С 2 -С 1 )-2/2 of the odd numbers ≤ maximal length of the alternance, (maximal amount of NOT residue of mod(Р n #)).
In the scientific works [7 pp. 142-147, 8 pp. 77-84, 9 pp.109-116], the distribution of groups of 4 consequent residues in the form of "pairs of residues every two residue" is analyzed. But we have no information on distribution of groups of 4, 3, and 2 consequent residues of mod(Р n #) for every fractal Р n #.
In this scientific work, the φ(Р n #) of the least residue of mod(Р n #) of every recurrent fractal -Р n # is indexed as continuous sequence of groups: (а) No 4 has got 4 residues, or (b) No 3 has got 3 residues or (с) No 2 has got 2 consequent residues mod(Р n #). These groups No 4-3-2 are analyzed as subgroups with No 4-3-2 consequent residues of mod(Р n #) that are surrounded by the maximal permissible amount of consequent NOT residues of mod(Р n #).
We used the mathematical induction method to define the overallmaximal length of every kind of subgroups No 4, No 3, No 2 and overall maximal length of P.  I. sections in the form of maximal long alternances of all first primes ≤Р n , (that is maximal permissible amount of all NOT residues of mod(Р n #)), situated between two residues from С А to С В , between which, as subgroups are situated the groups of residues of mod (Р n #). Type: As a result, we detected the loopback of these groups rearrangement from No 4 to No 3 up to No 2 according to the growing amount of the modulus, and the primes order distribution is defined.
And so on, repeating of fractal =5# with the period =5#.  With the unknown to us, length of the group No 2 from A P to B P and from С 1 to С 2 , for the mod(Р 1 #), is (R 2 -2)/2 of the odd numbers with: R 2 =( B P-A P)., Herewith for each group No 4-3-2 according to mod(Р 1 #), there are two residues of mod(Р 1 #): С А -to the left and С В -to the right, that is every group No 4-3-2 is the subgroup on the P.I. sections of the length unknown to us from С А to С В :

Correlations of length limits of the subgroups No 4, No 3, No 2
In the scientific works [7 pp. 142-147, 8 pp. 77-84, 9 p. 109-116, 10 p. 1805], including Section 5 of this work, the overallmaximal length of the subgroup No 4 (containing 4 residual for every recurrent fractal -Р n #), of type is defined: max R 4 =(С 4 -С 1 )=2Р n+1 of whole numbers. Herewith, it is quite obvious and it is beyond argument that relations of limits, unknown to us of groups No 4-3-2 length according to the increasing modulus are indicated in Table 10

Distribution of prime number
Correlation of length limits of the subgroups in Table 10 and distribution of groups of the indexed residues No 4-3-2, in every respective fractal -Р n # according to the increasing modulus is defined by theorem 1. Theorem 1. The loopback of prime number distribution. Every prescribed prime number squared = (Р 2 ) 2 defines the distribution of all previous prime numbers < (Р 2 ) 2 , as all first prime numbers are less than every prescribed prime number squared = (P 2 ) 2 , are situated in the P.I. as part of fractal Р 1 #, where they are distributed by subgroups of (а), (b), (с) types.
a. φ(Р 1 #) of subgroups No 4 having pure FOUR consequent prime number ( A,B, , and further, from Р 2 2 to Р 1 # pure FOUR consequent residues of mod(Р 1 #) on all P.I. sections with length not exceeding 2Р 3 of whole numbers with length of every subgroup No 4 at every section is: In that case, these primes of fractal = Р 1 #, by loopback, are distributed by groups: 2 2 type. At the P.I. section from Р 1 to Р 2 2 and further, from Р 2 2 to Р 1 # pure THREE consequent residues of mod(Р 1 #) on all P.I. sections with length not exceeding 2Р 2 of whole numbers with length of every subgroup No 3 at every section is: 2 2 type. At the P.I. section from Р 1 to Р 2 2 and further, from Р 2 2 to Р 1 # pure TWO consequent residues of mod(Р 1 #) on all P. I. sections with length not exceeding 2Р 1 of whole numbers with length of every subgroup No 2 at every section is: With: A,B,C,D С are consequent residues of mod(Р 1 #) including the primes < (Р 2 ) 2 . R 4-3-2 is the remainder of the first and the last number of every group No 4-3-2 (of the fractal Р 1 #). Further, the length of the subgroup No 4-3-2 as the amount of odd numbers, restricted by every group from A Cto B-C-D C, from A P to B-C-D P are (R 4,3,2 -2)/2 odd numbers.
The order of groups (а), (b), (с) rearrangement according to the increasing modulus for visual clarity is indicated in Table 11.

Proof of section a of the Theorem 1
It is feasible that in P.I. using the first prime number ≤ Р n (NOT residues of mod (Р n #)), by the only single way, we can form the maximal long P.I. section as the maximal long alternancearray for the 1 least common factor > 1 from every NOT residue of mod(Р n #). That is that maximal amount of NOT residues of mod(Р n #), maximal long alternance ≤Р n .

4 5
Fractal from 1 to Р n # Composition and its repeating =mod(Р n #) Length of P.I. section which defines values of primes number of this fractal including on the P.I section: Length of every section for the group No 4 … and so on according to the increasing meanings of the modulus … Then for every recurrent prime number Р n = Р 1 at the P.I., formed as recurrent line symmetrical, primary-repeatable periodical fractal = Р 1 # or I.R.S. according to mod(Р 1 #), (see the first line of Table 1), the maximal long P.I. section, formed as the alternance of the all first primes number ≤ Р 1 , (NOT residues of mod(Р n #)), shall be situated within the P.I. section from С А to С В , with as the subgroup is the only maximal long maximal subgroup No 4 (С 1 -С 2 -С 3 -С 4 ) with the 4 consequent residue of mod(Р 1 #): Type: The length of such maximally long subgroup No 4 of mod(Р 1 #), is: max The limit of length of the P.I. section within which from С А to С В would be situated maximal as well as all the other φ(Р 1 #) subgroups No 4 (with 4 residues) of mod(Р 1 #), is: It is genuinely: At the line-symmetrical, primary-repeatable fractal-Р 1 # or I.R.S. according to mod(Р 1 #), φ(Р 1 #) of the least residue (indexed in the form of φ(Р 1 #) groups No 4 of mod(Р 1 #), with the alternances ≤Р 1 with different lengths), are situated linesymmetrically relating to the center of symmetry of the fractal-Р 1 #, of the number = (Р 1 #/2). That is they are situated reflecting in pairs and are formed by two different ways: (the left and the right sieve of Eratosthenes), to the left and to the right from the symmetry center of the fractal = Р 1 # of number = Р 1 #/2. To the rightfor the increasing values numbers of the P.I. from Р 1 #/2 to Р 1 # to the left for the decreasing values of the P.I. Р 1 #/2 up to 1.
And the section of P.I. fractal Р 1 # (I.R.S. of mod(Р 1 #) from С А to С В ), represented by alternance ≤Р 1 with using of all NOT residues of mod(Р 1 #), (according to the 1 the least > 1), with the subgroup is situated the only maximally long -maximal group No 4 with 4 residues of mod(Р 1 #) (С 1 -С 2 -С 3 -С 4 ). Type: Thus in the fractal-Р 1 #-I.R.S. of the mod(Р 1 #), there is only one maximally long subgroup No 4, situated within the maximally long alternance ≤Р 1 , using all NOT residues of mod(Р 1 #), at the P.I. section ( It is quite obvious that all the other, line-symmetrical subgroups No 4 of mod (Р 1 #), situated within the alternances ≤Р 1 with different lengths or NOT residues, of mod(Р 1 #), cannot have the maximal length as they are formed by two different ways, that is they would be shorter than R 4 < 2P 2 , and situated within the P.I. sections from С А to С В with length not exceeding the maximal long P.I. section (С В -С А ) ≤2Р 3 of the whole numbers, not exceeding (Р 3 -1) of the odd numbers.
And so on, for all posterior prime numbers = Р n , at the increasing fractals -Р n # with n -as the whole number and proves the reality of the values of column No 3 of Table 11 and item (а) of Theorem 1.
It is feasible that there is such a prime number Р n = Р (1) , for which the P.I. is the line-symmetrical fractal Р (1) #, situated at the P.I. section from 1 to Р (1) # with subgroup No 4 (containing 4 residues) of mod(Р (1) #) with length = R 4 > 2*Р (2) , situated within the alternance of all first primes number ≤ Р (1) within the P.I. section with length > 2*Р (3) , >(Р (3) -1) of the odd numbers. Then every subgroup No 4 would be line-symmetrical to the left and to the right from the center of the fractal Р (1) # symmetry of the number Р (1) #/2. That is, in the result, we'll get in fractal Р (1) # using all primes number ≤ Р (1) , À we can by more than by one way from the maximally long alternance of all the prime numbers ≤ Р (1) , that is by the sieve of Eratosthenes, focused to the left and to the right (to the left and to the right from the number = Р (1) #/2), that is contrary to the taken axiom.
Order, type, and formula of indexing of two subgroups No 3 according to the increasing modulus are represented in Table 12 а,b,с.
And so on, for every of all posterior primes = Р n , at the increasing fractals Р n #, with n is the whole, represented in Tables 11 and 14 (the proof is indicated in Section 10). (Numerical illustrations are in Table 15).
As any eliminated residue С 2 , during the rearrangement of groups from No 3 to No 2 for the mod(Р 3 #) cannot change the length on no subgroup that is all R 2 would permanently ≤2Р 2 , and included in φ(Р 2 #) groups No 3 of mod(Р 2 #) there are two uncial, repeated Р 3 times maximally long subgroups No 3 of mod(Р 2 #) with the alternance ≤Р 2 with length maximal R 3 =2Р 2 , that would be rearranged into two maximally long line-symmetrical groups No 2 of mod(Р 3 #) by "eliminating" the residues С 2 with the number multiple Р 3 , (1 time in Р 3 lines). As all the other φ(Р 2 #)-2 subgroups, rearranged from No 3 to No 2 for the mod(Р 3 #), are shorter than (Р 2 -1) of the off numbers, that is: R 2 < 2Р 2 , and in Sections 9.1 and 9.3, it is proved that there are no other ways of comparing or rearranging of the subgroups No 2 of mod(Р 2 #) with length R 3 > 2Р 2 . The order, type, and formula of indexing of two subgroups No 2 according to the increasing modulus are represented in Table 13 b, с,d.
Herewith, it is quite obvious, and proved in Section 9.3, that all the other subgroups No 2 of mod(Р 3 #), with different lengths, rearranges from groups No 3 would be within the P.I., with length not exceeding the limit =2Р 3 of the wholes.
And so on, for every of all posterior primes = Р n , at the increasing fractals Р n #, with n is the whole, represented in Tables 11 and 14 (the proof is indicated in Section 10) (numerical illustrations are in Table 16).
8.2 Herewith, it is quite obvious that every three consequent residues of every subgroup No 3 according to the increasing group of mod(Р n #), represented in Table 17, are still within the P.I. section with length not exceeding -2Р n+1 of the whole numbers, as "eliminated" residues С 2;3 and С 1;4 of mod(Р n #) doesn't change the location of every subgroup No 3. That is, we'll get at the three adjoined groups No 3 lines of Table 17 for the mod( Including pure two subgroups No 3 max R 3 =2Р 2 , located within the maximally long section with length = 2Р 3 of the whole numbers. (The rearrangement is studied in Section 6). 8.3 Within the Р 2 duplications φ(Р 1 #) of the groups No 3 of mod(Р 1 #), number Р 2 "eliminated," that is rearranges to the mod(Р 2 #) 1 time every of all previously eliminated numbers of group No 3, except three residues С 1 -С 2 -С 3 . That is, transits to the mod(Р 2 #) (Р 2 -3) of No 3 groups in every φ(Р 1 #) column of No 3 groups.
And so on, for every of all eventual primes = Р n , represented as the loopback of groups distribution of the residues No 3 at the increasing fractals -Р n # according to the increasing meanings of modulus-mod(Р n #) that proves the validity of section (b) of the Theorem 1 (loopback of groups No 3 is represented in column No 3 of Table 14).

The loopback of rearrangement for φ(Р n #) groups No 2 (2 residues) from the mod(Р n-1 #) to mod(Р n #)
The looped back order of rearrangement φ(Р n #) of No 2 groups, according to the increasing modulus, that are represented in column No 4 of Table 14, in Section 9 are examined by steps for every recurrent increasing fractal -Р n #: Representing the first Р 2 lines in Table 1 as one line, we'll get the fractal -Р 2 # according to mod(Р 2 #)-I.R.S of mod(Р 2 #) at the P.I. sections from 1 to Р 2 # (1 line of Table 2).
Including pure two subgroups No 2 max R 2 =2Р 2 , located within the maximally long section with length = 2Р 3 of the whole numbers, two rearrangement is studies in Section 7.
And so on, for every from all eventual primes = Р n , in the form of loopback of residues of groups No 2 distribution at the increasing fractals -Р n #, with the increasing values of modulus of mod(Р n #), that proofs the validity of section (с)of Theorem 1 (loopback of groups No 2 is represented in column No 4 of Table 14).
Genuinely: Every P.I. section with length = 2 ffiffiffiffi N p of the whole numbers is located at the fractal -Р 1 # at the P.I. section. <Р 2 2 as, Р 1 2 < (N + 2 ffiffiffiffi N p Þ ≤ Р 2 2 , with: 2 ffiffiffiffi N p > 2Р 1 . It is feasible, that there is a P.I. section with length = 2 ffiffiffiffi N p of the whole numbers, where there are no two primes, that is, two consequent primes are located at the P.I. section with length exceeding -2 ffiffiffiffi N p of the whole numbers >2Р 1 , but this contradicts to section (с) of the Theorem "Loopback of primes distribution," that states, that is every fractal -Р 1 # according to mod(Р 1 #), on every P.I. sections with length not exceeding 2Р 1 of the whole numbers, there is a subgroup No 2 with 2 residues of mod(Р 1 #), that is two primes <Р 2 2 .

Author details
Shcherbakov Aleksandr Gennadiyevich Independent Researcher, Russia *Address all correspondence to: ag_ask@mail.ru © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.