Prime Numbers Distribution Line

Abstract

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 + 2N: (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 2N integers, that is, Р3–Р1 < 2N 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 2Nintegers.

1. Introduction

1.1 Line-symmetrical primary-repeatable fractals of the positive integers

In the scientific works [7 pp. 142–147, 8 pp. 77–84, 9 pp. 109–116], positive integers are analyzed, hereinafter the P.I. is represented only as the alternance (array) of primes (according to the 1st least prime factor > 1 from every whole number). Type: 12.3.5.7.3.11.13.3.17.19.3.23.5.3.29…3.р.р.3.р…3.р.р.3.р…. with for every recurrent prime = Р₁, sieve of Eratosthenes formats the P.I., represented by alternance (array) of the first primes ≤Р₁, in the form line-symmetrical repeating fractal-like structure, situated in the section of P.I. from 1 to Р₁#, with “eliminated” sections of P.I. and φ(Р₁#) not eliminated odd numbers are line-symmetrical to the number = Р₁#/2 and are repeated without rearrangement of their position with the period = Р₁#, on the basis of rhythmical repeating of two even numbers. Every recurrent prime has its own line-symmetrical primary-repeatable fractal = Р₁, then goes fractal = Р₁# (see line 1 Table 1).

С1 = 1	3,5,7..	С2 = Р2	pрр	..С3..	ррр	.. Сn ..	ррр	Р1#–Сn	ррр	Р1#–С2	..7,5.3	(Р1#–1)
1 + Р1#	3,5,7..	С2 + Р1#	pрр	С3 + Р1#	ррр	Сn + Р1#	ррр	2Р1#–Сn	ррр	2Р1#–С2	..7,5.3	(2Р1#–1)
1 + 2Р1#	3,5,7..	С2 + 2Р1#	pрр	С3 + 2Р1#	ррр	Сn + 2Р1#	ррр	3Р1#–Сn	ррр	3Р1#–С2	..7,5.3	(3Р1#–1)
…	3,5,7..	…	pрр	…	ррр	…	ррр	…	ррр	…	..7,5.3	…
…	3,5,7..	…	pрр	…	ррр	…	ррр	Р2#–Сn	ррр	Р2#–С2	..7,5.3	(Р2#–1)
1 + Р2#	3,5,7..	С2 + Р2#	pрр	С3 + Р2#	ррр	Сn + Р2#	ррр		ррр		..7,5.3
And so on, repeating of fractal = Р1# with the period = Р1#, with: ррр is alternance of ≤ Р1

Table 1.

Р₂ repeating of periodical fractal = Р₁#, including I.R.S. according to the mod(Р₁#).

С1 = 1	3,5,7,,,	С2 = Р3	pрр	..С3..	ррр	.. Сn ..	ррр	Р2#–Сn	ррр	Р2#–С2	..7,5.3	(Р2#–1)
1 + Р2#	3,5,7,,,	С2 + Р2#	pрр	С3 + Р2#	ррр	Сn + Р2#	ррр	2Р2#–Сn	ррр	2Р2#–С2	..7,5.3	(2Р2#–1)
1 + 2Р2#	3,5,7,,,	С2 + 2Р2#	pрр	С3 + 2Р2#	ррр	Сn + 2Р2#	ррр	3Р2#–Сn	ррр	3Р2#–С2	..7,5.3	(3Р2#–1)
…	3,5,7,,,	…	pрр	…	ррр	…	ррр	…	ррр	…	..7,5.3	…
…	3,5,7,,,	…	pрр	…	ррр	…	ррр	Р3#–Сn	ррр	Р3#–С2	..7,5.3	(Р3#–1)
1 + Р3#	3,5,7,,,	С2 + Р3#	pрр	С3 + Р3#	ррр	Сn + Р3#	ррр	…	ррр	…	..7,5.3	…
And so on, repeating of fractal = Р2# with the period = Р2#, with: ррр is alternance of ≤ Р2.; Representing the section of P.I. as line from 1 to Р3# we’ll get the fractal Р3# and so on.

Table 2.

Р₃ repeating of periodical fractal = Р₂#, including I.R.S. according to the mod(Р₂#).

Every recurrent line-symmetrical fractal -Р₁# is situated on the section of P.I. from 1 to Р₁# and contains φ(Р₁#) of the not eliminated odd numbers that are φ(Р₁#) of the least residue, belonging to the indicated residue system (I.R.S) according to mod (Р₁#), type: С_n to the left from the number = Р₁#/2 and (Р₁#–С_n) to the right from the number = Р₁#/2, with С_n – is residue according to mod(Р₁#). Hereinafter with the term mod(Р₁#), we shall indicate the period of fractal Р₁# repetition (I.R.S, sieve of Eratosthenes), equal to product of all first primes ≤Р₁ (primorial = Р₁#) [1, 2, 3, 4, 5, 6].

By term residue according to mod (Р₁#), we shall indicate every number, NOT eliminated by Sieve of Eratosthenes, not aliquot to the first primes ≤Р₁.

Alternance ≤Р₁ is the section of P.I. in the form of array of primes – NOT residues of mod(Р₁#), (for the 1 least common factor > 1 from every NOT residue).

Eliminating (according to diagonals) 1 number multiple to Р₂ in every column = Сn, we’ll get in Р₂ lines of Table 1: φ(Р₁#)*(Р₂ lines)─φ(Р₁#)_multiple Р₂ = φ(Р₂#) residue of mod(Р₂#). Representing the section of P.I. from 1 to Р₂# as one line, we’ll get the fractal = Р₂# with the period of repeating =Р₂#. And so on: every recurrent prime = Р_n has got its periodical fractal = Р_n# with n is the whole. The numerical illustration is indicated in the scientific works [7, 8, 9, 10].

Fractal (Р₁#)-I.R.S. mod(Р₁#) = (first line of Table 1).

Fractal (Р₂#)-I.R.S. mod(Р₂#) = Р₂ lines in Table 1. -φ(Р₁#) numbers multiple to Р₂.

Fractal (Р₃#) I.R.S. mod(Р₃#) = Р₃ lines in Table 2.-φ(Р₂#) numbers multiple to Р₃.

Fractal (Р₄#) I.R.S. mod(Р₄#) = (Р₄ repeating of fractal Р₃#)--φ(Р₃#) numbers multiple to Р₄.

Fractal (Р₅#) I.R.S. mod(Р₅#) = (Р₅ repeating of fractal Р₄#)--φ(Р₄#) numbers multiple to Р₅.

and so on according to cumulative primes.

1.2 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.С3рр3С., 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 С₁…3рр3рр3рр3…С₂ 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: (С₂ –С₁)–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 overall – maximal 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:

1. -No 4: С_А..3рр3..Р_1..Р_2..Р_3..Р_4..3рр3..С_В.

2. No 3: С_А..3рр3..Р_1..Р_2..Р_3..3рр3..С_В.

3. No 2: С_{А..3 рр3..}Р_1..Р_2..3рр3..С_В.

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.

2. Three groups of “eliminated” sections of every next fractal

It is quite obvious and requires no proof that indexing φ(Р₁#) of the least residues of mod(Р₁#) of every recurrent fractal-Р₁#, or I.R.S mod(Р₁#), is by groups, containing strictly 4 elements; three; two consequent residues of mod(Р₁#), we have, that every recurrent fractal-Р₁# would be represented as array of three groups of the residues of mod(Р₁#), between are situated the alternances ≤Р₁ (with different lengths) – the consequent NOT residues of mod(Р₁#), of types (а), (b), (с) repeated without changes with period = Р₁#.

1,3,5,7…	AP ..3рр3..BP..3рр3..CP..3рр3..DP. 3рр3…	P2^2	1С.3рр3.4С.3рр3..	Р1#
3,5,7…	-------BP..3рр3..CP..3рр3..DP..3рр3..AP…	P2^2	---2С..3рр3..5С..2C	Р1#
5,7…	- -----CP..3рр3..DP.3рр3..AP..3рр3..BP	P2^2	-----3С..3рр3..6С..3C	Р1#
And so on, repeating of fractal = Р1# with the period = Р1#, ррр - is alternance of ≤ Р1.

Table 3.

Fractal = Р₁#, represented as φ(Р₁#) of No 4 groups (containing 4 residues) of mod(Р₁#). Three adjoined groups No 4 for every residue = С.

1. φ(Р₁#) groups No 4 containing strictly FOUR consequent residues of mod(Р₁#)_A,B,C,DС, between which the alternances of different amounts of different first primes ≤Р₁, NOT residues of mod(Р₁#), type: _AС..3рр3.._BС..3рр3.._CС .. 3рр3.._DС. with: ..3рр3.. are alternances of the first primes ≤Р₁ according to the 1st least common factor > 1 from every NOT residue of mod(Р₁#). (_1–4С) – Four consequent residue of mod(Р₁#), including the consequent primes of P.I. section from Р₁ to (Р₂)² of _A,B,C,DР type. Further, the fractal = Р₁# represented as φ(Р₁#) of No 4 groups (4 residues) of mod(Р₁#). Three adjoined groups No 4 for every residue = С (Table 3).

   The length of group No 4, which means amount odd numbers, restricted by every group No 4 from _AP to _DP and from С₁ to С₄, for the mod(Р₁#), is (R₄–2)/2 ≤ (Р₂–1) of the odd numbers with: R₄ = (_DP--_AP), R₄ = (₄С–₁С), R₄ ≤ 2Р₂ (including 1 group of R₄ = 2Р₂, detailed information is indicated in Section 5) (Table 4).

2. φ(Р₁#) groups No 3 containing strictly THREE consequent residues of mod(Р₁#). _A,B,CС, between which the alternances of different amounts of different first prime ≤Р₁, NOT residues of mod (Р₁#), type: _AС..3рр3.._BС..3рр3.._CС. with: ..3рр3.. are alternances of the first primes ≤Р₁ according to the 1st least common factor > 1 from every NOT residue of mod(Р₁#). (_1-3С) – three consequent residues of mod(Р₁#), including the consequent primes of P.I. section from Р₁ to (Р₂)² of _A,B,CР type. Further, the fractal = Р₁# represented as φ(Р₁#) of No 3 groups (3 residues) of mod(Р₁#). Two adjoined groups No 3 for every residue = С (Table 5).

   With the unknown to us length of the group No 3 from _AP to _CP and from С₁ to С₃, for the mod (Р₁#), is (R₃–2)/2 of the odd numbers with: R₃ = (_CP--_AP)., R₃ = (₃С–₁С)., R₃ =? (it is quite obvious that for mod (Р₁#) R₃ < <R₄) (Table 6).

3. φ(Р₁#) groups No 2 containing strictly TWO consequent residues of mod(Р₁#)_A,BС, between which the alternances of different amounts of different first prime ≤Р₁, NOT residues of mod(Р₁#), type: _AС..3рр3.._BС. with: ..3рр3.. are alternances of the first primes ≤Р₁ according to the 1st least common factor > 1 from every NOT residue of mod(Р₁#). (_1–2С) – two consequent residue of mod(Р₁#), including the consequent primes of P.I. section from Р₁ to (Р₂)² of _A,BР type. Further, the fractal = Р₁# represented as φ(Р₁#) of No 2 groups (2 residues) of mod(Р₁#) (Tables 7–9).

Type-(a)	С1 = 1	7	11	С4 = 13	17	19	С7 = 23	29	31	С = 37
Type-(b)		С2 = 7	11	13	С5 = 17	19	23	С8 = 29	31	37	С = 41
Type-(c)			С3 = 11	13	17	С6 = 19	23	29	С = 31
And so on, repeating of fractal =5# with the period = mod(5#)

Table 4.

The numerical illustration of the fractal =5# in the form of φ(5#) = 8 groups No 4 (containing four residues). The three adjoined groups No 4 for every residue = С with R₄ ≤ 2Р₂ = 2*7 (consult Section 5).

1.,3.,5.,7..	AР..3рр3..BP ..3рр3..CP	P2^2	1С..3рр3..3С…3рр3.. 1С…3С	Р1#
3.,5.,7…	--------.BP ..3рр3..CP..3рр3..АP..3рр3..	P2^2	--------2С..3рр3..4С.. 2С .. 4С	Р1#
And so on, repeating of fractal = Р1# with the period = Р1#, with: ррр is alternance of ≤ Р1.

Table 5.

Fractal = Р₁#, represented as φ(Р₁#) of No 3 groups (containing 3 residues) of mod(Р₁#). Two adjoined groups No 3 for every residue = С.

Type-(a)	С1 = 1	7	С3 = 11	13	С5 = 17	19	С7 = 23	29	С =31
Type-(b)		С2 = 7	11	С4 = 13	17	С6 = 19	23	С8 = 29	31	С = 37
And so on, repeating of fractal =5# with the period =5#.

Table 6.

The numerical illustration of the fractal =5# in the form of φ(5#) = 8 groups No 3 (containing two residues). The two adjoined groups No 3 for every residue = С with R₃ =? (=2*5 consult Section 6).

1.,3.,5.,7..	AР..3рр3..BP..3рр3..АP.3рр3..	P2^2	1С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С…	Р1#
And so on, repeating of fractal = Р1# with the period = Р1#, ррр - is alternance of≤ Р1.

Table 7.

Fractal = Р₁#, represented as φ(Р₁#) of No 2 groups of mod(Р₁#).

1.,3.,5.,7..	AР..3рр3..BP..3рр3..АP.3рр3..	P3^2	1С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С…	Р2#
And so on, repeating of fractal = Р2# with the period = Р2#, ррр - is alternance of ≤ Р2.

Table 8.

Fractal = Р₂#, represented as φ(Р₂#) of No 2 groups of mod(Р₂#).

1.,3.,5.,7..	AР..3рр3..BP..3рр3..АP.3рр3..	P4^2	1С..3рр3..2С..3рр3..1С..3рр3..2С..1С..2С…	Р3#
And so on, repeating of fractal = Р3# with the period = Р3#, ррр - is alternance of ≤Р3.

Table 9.

Fractal = Р₃#, represented as φ(Р₃#) of No 2 groups of mod(Р₃#).

With the unknown to us, length of the group No 2 from _AP to _BP and from С₁ to С₂, for the mod(Р₁#), is (R₂–2)/2 of the odd numbers with: R₂ = (_BP--_AP)., R₂ = (₂С–₁С), R₂ =? (it is quite obvious that for mod(Р₁#) R₂ <<R₃).

Herewith for each group No 4-3-2 according to mod(Р₁#), there are two residues of mod(Р₁#): 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: (а)С_А-(С₁-С₂-С₃-С₄)-С_В. (b) С_А-(С₁-С₂-С₃)-С_В. (с) С_А-(С₁-С₂)-С_В.

3. 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 overall – maximal length of the subgroup No 4 (containing 4 residual for every recurrent fractal -Р_n#), of type is defined:

max R₄ = (С₄ – С₁) = 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.

Prime value Рn	Fractal; - Рn#	Period of fractal repetition; =mod(Рn#).	Max length of φ(Рn#) of the subgroups No 4; maxR4 = (С4–С1)	>>	Max length; φ(Рn#) of the subgroups No 3; maxR3 = (С3–С1)	>>	Max length; φ(Рn#) of the subgroups No 2; maxR2 = (С2–С1)
Р1	Р1#	mod(Р1#).	max R4 =; (С4–С1) = 2Р2	>>	max R3 =; (С3–С1) =?	>>	max R2 =; (С2–С1) =?
Р2	Р2#	mod(Р2#).	max R4 =; (С4–С1) = 2Р3	>>	max R3 =; (С3–С1) =?	>>	max R2 =; (С2–С1) =?
Р4	Р4#	mod(Р4#).	max R4 =; (С4–С1) = 2Р5	>>	max R3 =; (С3–С1) =?	>>	max R2 =; (С2–С1) =?
Р5	Р5#	mod(Р5#).	max R4 =; (С4–С1) = 2Р6	>>	max R3 =; (С3–С1) =?	>>	max R2 =; (С2–С1) =?
…	…	…	…	>>	…	>>	…
Рn	Рn#	mod(Рn#).	max R4 =; (С4–С1) = 2Рn + 1	>>	max R3 =; (С3–С1) =?	>>	max R2=; (С2–С1) =?
And so on, repeating of fractal = Рn# with the period = Рn#. C1–4 - residue of mod (Рn#).

Table 10.

The relation of length limits of the subgroups according to the increasing modulus.

4. 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 = (Р₂)² defines the distribution of all previous prime numbers < (Р₂)², as all first prime numbers are less than every prescribed prime number squared = (P₂)², are situated in the P.I. as part of fractal Р₁#, where they are distributed by subgroups of (а), (b), (с) types.

1. φ(Р₁#) of subgroups No 4 having pure FOUR consequent prime number (_A,B,C,DС) of P₁ < (_AС-_BС-_CС-_DС) < P₂²; Р₃² type. At the P.I. section from Р₁ to Р₂² (including from (Р₂–2)² to Р₂²), and further, from Р₂² to Р₁# pure FOUR consequent residues of mod(Р₁#) on all P.I. sections with length not exceeding 2Р₃ of whole numbers with length of every subgroup No 4 at every section is:

   $${\text{R}}_4=\left({}_{\text{D}}{\text{С–}}_{\text{A}}\text{С}\right)\le 2{\text{Р}}_2.$$

   In that case, these primes of fractal = Р₁#, by loopback, are distributed by groups:

2. φ(Р₁#) of subgroups No 3 having pure THREE consequent prime number (_A,B,CС) of P₁ < (_AС-_BС-_CС) < P₂² type. At the P.I. section from Р₁ to Р₂² and further, from Р₂² to Р₁# pure THREE consequent residues of mod(Р₁#) on all P.I. sections with length not exceeding 2Р₂ of whole numbers with length of every subgroup No 3 at every section is: R₃ = (_CС–_AС) ≤ 2Р₁.

3. φ(Р₁#) of subgroups No 2 having pure TWO consequent prime number (_A,BС) of P₀, P₁ < (_AС-_BС) < P₂² type. At the P.I. section from Р₁ to Р₂² and further, from Р₂² to Р₁# pure TWO consequent residues of mod(Р₁#) on all P.I. sections with length not exceeding 2Р₁ of whole numbers with length of every subgroup No 2 at every section is: R₂ = (_BС–_AС) ≤ 2Р₀.

With: _A,B,C,DС are consequent residues of mod(Р₁#) including the primes < (Р₂)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 Р₁#). Further, the length of the subgroup No 4-3-2 as the amount of odd numbers, restricted by every group from _AC to _B-C-DC, from _AP to _B-C-DP 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.

1	2		3		4		5
Fractal from 1 to Рn#; Composition and its repeating =mod(Рn#)	Length of P.I. section which defines values of primes number of this fractal including on the P.I section: (Рn + 1)^2–(Рn + 1–2)^2 = 4Рn + 1 of whole numbers		φ(Рn#) groups No 4 for 4 residues of mod(Рn#) (containing 4 simple) AС-BС-CС-DС < Pn + 1^2		φ(Рn#) groups No 3 for 3 residues of mod(Рn#) (containing 3 simple) AС-BС-CС < Pn + 1^2		φ(Рn#) groups No 2 for 2 residues of mod(Рn#) (containing 2 simple) (AС-BС) < Pn + 1^2
			Subgroup No 4 length. R4 = DС--AС	Length of every section for the group No 4	Subgroup No 3 length. R3 = CС--AС	Length of every section for the group No 3	Subgroup No 2 length. R2 = BС--AС	Length of every section for the group No 2
1 ÷ Рn# mod(Рn#)	from (Рn)^2 to (Рn + 1)^2	≥4Рn + 1 number	R4 ≤ 2Pn + 1; <(Pn# ± Pn + 1)	≤2Pn + 2 <(Pn# ± Pn + 2)	R3 ≤ 2Pn; (Table 12)	≤2Pn + 1 number	R2 ≤ 2Pn–1 (Table 13)	≤2Pn number
…	…	…	…	…	…	…	…	…
1 ÷ Р1# mod(Р1#)	from (Р1)^2 to (Р2)^2	≥4Р2 number	R4 ≤ 2P2	≤2P3 number	R3 ≤ 2P1	≤2P2 number	R2 ≤ 2P0	≤2P1 number
1 ÷ Р2# mod(Р2#)	from (Р2)^2 to (Р3)^2	≥4Р3 number	R4 ≤ 2P3	≤2P4 number	R3 ≤ 2P2	≤2P3 number	R2 ≤ 2P1	≤2P2 number
1 ÷ Р3# mod(Р3#)	from (Р3)^2 to (Р4) ^2	≥4Р4 number	R4 ≤ 2P4	≤2P5 number	R3 ≤ 2P3	≤2P4 number	R2 ≤ 2P2	≤2P3 number
1 ÷ Р4# mod(Р4#)	from (Р4)^2 to (Р5) ^2	≥4Р5 number	R4 ≤ 2P5	≤2P6 number	R3 ≤ 2P4	≤2P5 number	R2 ≤ 2P3	≤2P4 number
1 ÷ Р5# mod(Р5#)	from (Р5)^2 to (Р6) ^2	≥4Р6 number	R4 ≤ 2P6	≤2P7 number	R3 ≤ 2P5	≤2P6 number	R2 ≤ 2P4	≤2P5 number
…and so on according to the increasing meanings of the modulus…

Table 11.

Loopback of primes number subgroups distribution according to the increasing meanings of the modulus.

5. Proof of theorem

5.1 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 alternance – array 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.

Then for every recurrent prime number Р_n = Р₁ at the P.I., formed as recurrent line symmetrical, primary-repeatable periodical fractal = Р₁# or I.R.S. according to mod(Р₁#), (see the first line of Table 1), the maximal long P.I. section, formed as the alternance of the all first primes number ≤ Р₁, (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 (С₁-С₂-С₃-С₄) with the 4 consequent residue of mod(Р₁#): Type: С_А = (Р₁#--Р₃)..С₁ = (Р₁#--Р₂).

…С₂ = (Р₁#--1),С₃ = (Р₁# + 1) … С₄ = (Р₁# + Р₂)… С_В = (Р₁# + Р₃). The length of such maximally long subgroup No 4 of mod(Р₁#), is: max R₄ = (С₄–С₁) = (Р₁# + Р₂)--(Р₁#--Р₂) = 2Р₂ of the whole numbers.

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(Р₁#) subgroups No 4 (with 4 residues) of mod(Р₁#), is: (С_В–С_А) = (Р₁# + Р₃)–(Р₁#–Р₃) = 2Р₃ of the whole numbers.

It is genuinely:

At the line-symmetrical, primary-repeatable fractal-Р₁# or I.R.S. according to mod(Р₁#), φ(Р₁#) of the least residue (indexed in the form of φ(Р₁#) groups No 4 of mod(Р₁#), with the alternances ≤Р₁ with different lengths), are situated line-symmetrically relating to the center of symmetry of the fractal-Р₁#, of the number = (Р₁#/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 = Р₁# of number = Р₁#/2. To the right – for the increasing values numbers of the P.I. from Р₁#/2 to Р₁# to the left for the decreasing values of the P.I. Р₁#/2 up to 1.

To every left group No 4 of mod(Р₁#), with the remainder R₄ = (С₄–С₁), is matched by line-symmetrical right group No 4 mod(Р₁#), with reflecting location of the same first primes number in the same amount and of the same length of the alternance ≤Р₁: R₄ = (Р₁#–С₁)--(Р₁#–С₄) = (С₄–С₁) consult [7 pp. 142–147, 8 pp. 77–84, 9 pp. 109–116].

Besides two not reflecting that is formed solely subgroup of group No 4: with constant reminder for every Р_n of type: R₄ = (Р₁#/2 + 4)--(Р₁#/2–4) = 8.

And the section of P.I. fractal Р₁# (I.R.S. of mod(Р₁#) from С_А to С_В), represented by alternance ≤Р₁ with using of all NOT residues of mod(Р₁#), (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(Р₁#) (С₁-С₂-С₃-С₄). Type: С_А = (Р₁#-Р₃)_…3рр3… С₁ = (Р₁#-Р₂)_…3рр3…С₂ = (Р₁#-1), С₃ = (Р₁# + 1)_…3рр3…С₄ = (Р₁# + Р₂)_{… …3рр3…}С_В = (Р₁# + Р₃).

Thus in the fractal-Р₁#-I.R.S. of the mod(Р₁#), there is only one maximally long subgroup No 4, situated within the maximally long alternance ≤Р₁, using all NOT residues of mod(Р₁#), at the P.I. section (Р₁# ± Р₂) with length maximal R₄ = 2P₂ restricting (R₄–2)/2 = (2P₂--2)/2 = (Р₂--1) of the odd numbers, situated within the P.I. section, formed solely from (Р₁#--Р₃) to (Р₁# + Р₃) with length of (Р₃--1) of odd numbers.

It is quite obvious that all the other, line-symmetrical subgroups No 4 of mod (Р₁#), situated within the alternances ≤Р₁ with different lengths or NOT residues, of mod(Р₁#), cannot have the maximal length as they are formed by two different ways, that is they would be shorter than R₄ < 2P₂, 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) ≤2Р₃ of the whole numbers, not exceeding (Р₃–1) 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 = R₄ > 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₎, − 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.

6. The maximal length of P.I. section with maximal long subgroups No 3 (with 3 residues) for the mod(Р₂#)

At the fractal -Р₁#, there are φ(Р₁#) subgroups No 4 of mod(Р₁#), with length R₄ ≤ 2Р₂ of the whole numbers including one maximal long subgroup No 4 of mod(Р₁#) with length max R₄ = 2Р₂ of the whole numbers. At the transition from mod (Р₁#) to mod(Р₂#), the fractal Р₁# and φ(Р₁#) of groups No 4 repeat Р₂ times. Then at the P.I. section from 1 to Р₂# (at Р₂ lines of Table 1), we’ll get φ(Р₁#) columns of groups No 4 of mod(Р₁#), with length R₄ ≤ 2Р₂, (Р₂ lines at the column No 4).

It is quite obvious that in Section 8.1, it is proved that if by number Р₂, “eliminate,” that is moved to mod(Р₂#) 1 time every elimination С₂ and С₃, in the column of every φ(Р₁#) group No 4 of mod(Р₁#), than at the P.I. section from 1 to Р₂#, (that is at the fractal Р₂#), we’ll get =2φ(Р₁#) groups No 3 of mod(Р₂#) of the same length, that is R₄ ≤ 2Р₂ of mod(Р₁#) would become = R₃ ≤ 2Р₂ of mod(Р₂#) with changing of the alternances composition from ≤Р₁ to ≤Р₂.

As all one by one eliminated residues С₂ or С₃, at rearrangement of the groups from No 4 to No3 for the mod(Р₂#), cannot change the length of none of the subgroups, that is all R₃ would permanently be ≤2Р₂, included in 2φ(Р₁#) groups No 3 of mod (Р₂#) there is only one Р₂ times repeated, maximally long subgroup No 4 of mod(Р₁#) with the alternance ≤Р₁, with length max R₄ = 2Р₂, that would be restructured into two maximally long line-symmetrical groups No 3 of mod(Р₂#) by “eliminating” the residues С₂ and С₃ by number multiple to Р₂, (1 time at Р₂ lines). As all the other 2φ(Р₁#)--2 subgroups, changed from No 4 to No 3 for mod(Р₂#) are shorter than (Р₂–1) of the odd numbers that is: R₃ < 2Р₂. In Sections 8.1 and 8.3, there are no other ways of making or changing the subgroups No 3 of mod(Р₂#) with length R₃ > 2Р₂.

Order, type, and formula of indexing of two subgroups No 3 according to the increasing modulus are represented in Table 12 а, b, с.

(а) Fractal = Р2#, φ(Р2#) group No 3 of mod(Р2#), alternance ≤Р2, maxR3 = 2Р2, with n = (multiple Р2 ± 1)/Р1# = the whole. At the P.I. sections (nР1# ± Р2) and (Р2--n)Р1# ± Р2. Within the limits of P.I. section (nР1# ± Р3) with: n and (Р2–n) is line number in Table 1
….СА….; =nР1#--Р3; АС = Р2#--СВ; (Р2--n)Р1#--Р3	.; .	……С1…; =nР1#--Р2; 1С = Р2#--С3; (Р2--n)Р1#--Р2	11,; 3,; 7,; 5,; 3	(кр.Р2) и С2; =nР1#. ± .1; 2С и (кр.Р2); (Р2-n)Р1# ± 1	3,; 5,; 7,; 3,; 11.	……С3…; =nР1# + Р2; 3С = Р2#--С1; (Р2--n)Р1# + Р2	..	….СВ……; =nР1# + Р3; ВС = Р2#--СА; (Р2--n)Р1# + Р3
(b) Fractal = Р3#, φ(Р3#) group No 3 of mod(Р3#), alternance ≤Р3, maxR3 = 2Р3, with n = (multiple Р3 ± 1)/Р2# = the whole. At the P.I. sections (nР2# ± Р3) and (Р3--n)Р2# ± Р3.; Within the limits of P.I. section (nР2# ± Р4) with: n and (Р3–n) is line number on Table 2
…СА….; =nР2#--Р4; АС = Р3#--СВ; (Р3--n)Р2#--Р4	.; .	……С1…; =nР2#--Р3; 1С = Р2#--С3; (Р3--n)Р2#--Р3	11,; 3,; 7,; 5,; 3	(кр.Р3) и С2; =nР2#. ± .1; 2С и (кр.Р3); (Р3--n)Р2# ± 1	3,; 5,; 7,; 3,; 11,	……С3…; =nР2# + Р3; 3С = Р2#--С1; (Р3--n)Р2# + Р3	..	….СВ……; =nР2# + Р4; ВС = Р2#--СА; (Р3--n)Р2# + Р4
(с) And so on for every mod(Рn#), maxR3 = 2Рn, n = (кр.Рn ± 1)/Рn-1# = the whole, Рn-(1)-primes

Table 12.

Type and formula for indexing of two line-symmetrical, maximally long subgroups No 3 (having 3 residues) at the increasing fractal according to the increasing modulus (Tables 11 and 14).

(b) Fractal = Р3#, φ(Р3#) group No 2 of mod(Р3#), alternance ≤Р3, maxR2 = 2Р2, with n = (multiple Р2*Р3 ± 1)/Р1# = the whole. At the P.I. sections (nР1# ± Р2) and (Р2*Р3--n)Р1# ± Р2.; Within the limits of P.I. section (nР1# ± Р3)., with: n and (Р2*Р3–n) is line number on Table 1
……СА….; =nР1#--Р3; АС = Р3#--СВ; (Р2Р3--n)Р1#--Р3	.; .	……С1…; =nР1#--Р2; 1С = Р3#--С3; (Р2Р3--n)Р1#--Р2	..; ..; 7,; 5,; 3	(кр.Р2) (кр.Р3); =nР1#. ± .1; (кр.Р3) (кр.Р2); (Р2Р3--n)Р1# ± 1	3,; 5,; 7.; ..; ..	……С2…; =nР1# + Р2; 2С = Р3#--С1; (Р2Р3--n)Р1# + Р2	.; .	….СВ……; =nР1# + Р3; ВС = Р3#--СА; (Р2Р3--n)Р1# + Р3
(c) Fractal = Р4#, φ(Р4#) group No 2 of mod(Р4#), alternance ≤Р4, maxR2 = 2Р3, with n = (multiple Р4*Р3 ± 1)/Р2# = the whole. At the P.I. sections (nР2# ± Р3) and (Р4*Р3--n)Р2# ± Р3.; Within the limits of P.I. section (nР2# ± Р4)., with: n-and (Р4*Р3–n) is line number on Table 2
……СА….; =nР2#--Р4; АС = Р4#--СВ; (Р4Р3--n)Р2#--Р4	.; .	……С1…; =nР2#--Р3; 1С = Р4#--С3; (Р4Р3--n)Р2#--Р3	..; ..; 7,; 5,; 3	(кр.Р4) (кр.Р3); =nР2#. ± .1; (кр.Р3) (кр.Р4); (Р4Р3--n)Р2# ± 1	3,; 5,; 7.; ..; ..	……С2…; =nР2# + Р3; 2С = Р4#--С1; (Р4Р3--n)Р2# + Р3	.; .	….СВ……; =nР2# + Р4; ВС = Р4#--СА; (Р4Р3--n)Р2# + Р4
(d) And so on for every mod(Рn + 1#), maxR2 = 2Рn, n = (Рn + 1*Рn ± 1)/Рn-1# = the whole, Рn-primes

Table 13.

Type and formula for indexing of two line-symmetrical, maximally long subgroups No 2 (having 2 residues) at the increasing fractal according to the increasing modulus (Tables 11 and 14).

The length of these two line-symmetrical subgroups No 3 of mod(Р₂#), that is the length of alternance ≤Р₂ from С₁ to С₃, is maximal R₃ = (С₃--С₁) = (nР₁#. ± Р₂)--(nР₁#.--Р₂) =2Р₂; (Р₂–1) of the odd numbers. Two of these subgroups No 3 are situated within P.I. section (nР₁#. ± Р₃) with length (С_В–С_А) = 2Р₃; (Р₃–1) of the odd numbers from С_А to С_В. Numerical values of these two maximal subgroups No 3 are defined according to the formula (multiple Р₂ and С₂ = multiple Р₂ ± 2) is (nР₁#. ± .1) and (Р₂--n)Р₁# ± 1, with n and (Р₂–n) define the number of the line for the group No 3 of mod(Р₂#) in column Р₂ and the repeated maximal of the group No 4 of mod(Р₁#) with the period = Р₁# (consult Table 1). That is n = (multiple Р₂. ± 1)/Р₁# = the whole < Р₂/2.

Whereas, it is quite obvious that in and proved in Section 8.2, the other subgroups No 3 of mod(Р₂#), with different lengths, changed from groups No 4 would be within P.I. section, with limit length = 2Р₃ of the whole numbers.

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).

1	2	3					4
Fractal composition -Рn#	φ(Рn#) groups No 4 of mod (Рn#).; Including 1 group = maxR4; Proof; in Section 5	Loopback (Рn--3) φ(Рn-1#); groups No 3 + loopback; 2φ(Рn-1#) group No 4; [modРn-1#]=; = φ(Рn#) groups No 3 of mod(Рn#) including; 2 groups No 3; = max R3 of mod (Рn#); Proved in Sections 6 and 8					Loopback (Рn--2) φ(Рn-1#); Groups No 2 + loopback; φ(Рn-1#) groups No 3; [modРn-1#]=; =φ(Рn#) groups No 2 of mod(Рn#) including; 2 groups No 2; = max R2 mod (Рn#); Proved in Sections 7 and 9
By repeating Р1 times the line of fractal -Р0# and group No 4-3-2, we’ll get φ(Р0#) columns of groups No 4-3-2 of mod(Р0#) within the alternances ≤Р0 (Р1 line in the column).; By “eliminating” 1 number multiple -Р1 (in line of every column No 4-3-2), that is by transiting this group for mod(Р1#) with changing of its length from R(4,3) to R3,2 and alternances composition from ≤ Р0 to ≤ Р1. At the P.I. section from 1 to Р1# we’ll get the fractal -Р1# of mod(Р1#). φ(Р1#) groups No 4-3-2
Amount of groups of; Length and location	φ(Р1#) groups; No 4. R4. ≤ 2Р2 at every sectionе<2Р3 is group No 4	φ(Р0#)*; *(Р1--3); R(3) ≤ 2Р0	+	2φ(Р0#); R(4) ≤ 2Р1	=	φ(Р1#); group No 3; R3 ≤ 2Р1	φ(Р0#)*; *(Р1--2); R(2) ≤ 2Р..	+	φ(Р0#); R(3) ≤ 2Р0	=	φ(Р1#); group No 2; R2 ≤ 2Р0
		At every section <2Р2 is group No 3					At every section <2Р1 is group No 2
Formula = n; Type max R; Groups and location	(Р1# ± Р2); max R4 = 2Р2; At the section with length = (Р1# ± Р3)	n = (multiple Р1 ± 1) / Р0# = the whole; (nР0#. ± .Р1) and (Р1--n)Р0#. ± .Р1; 2 groups No 3- max R3 = 2Р1.; On the segment of length = (nР0# ± Р2)					n = (multiple Р0*Р1 ± 1)/Р(.)# = the whole; (nР(.)# ± Р0) and (Р0Р1--n)Р(.)# ± Р0; 2 groups No2- max R2 = 2Р0.; On the segment of length = (nР(..)# ± Р1)
By repeating Р2 times the line of fractal –Р1# and group No 4-3-2, we’ll get φ(Р1#) columns of groups No 4-3-2 of mod(Р1#) within the alternances ≤Р1 (Р2 line in the column).; By “eliminating” 1 number multiple -Р2 (in line of every column No 4-3-2), that is by transiting this group for mod(Р2#) with changing of its length from R(4,3) to R3,2 and alternances composition from ≤ Р1 to ≤ Р2. At the P.I. section from 1 to Р2# we’ll get the fractal –Р2# of mod(Р2#). φ(Р2#) groups No 4-3-2
Amount of groups of length and location	φ(Р2#) groups; No 4. R4. ≤ 2Р3 at every sectionе <; 2Р4 is group No 4	φ(Р1#)*; *(Р2--3); R(3) ≤ 2Р1	+	2φ(Р1#); R(4) ≤ 2Р2	=	φ(Р2#); group No 3; R3 ≤ 2Р2	φ(Р1#)*; *(Р2--2); R(2) ≤ 2Р0	+	φ(Р1#); R(3) ≤ 2Р1	=	φ(Р2#); group No 2; R2 ≤ 2Р1
		At every section <2Р3 is group No 3					At every section <2Р2 is group No 2
Formula = n; Type max R; Groups and location	(Р2# ± Р3); max R4 = 2Р3; At the section with length = (Р2# ± Р4)	n = (multiple Р2 ± 1) / Р1# = the whole; (nР1#. ± .Р2) and (Р2--n)Р1#. ± .Р2; 2 groups No 3- max R3 = 2Р2.; On the segment of length = (nР1# ± Р3)					n = (multiple Р2*Р1 ± 1)/Р0# = the whole; (nР0# ± Р1) and (Р2Р1--n)Р0# ± Р1; 2 groups No2- max R2 = 2Р1.; On the segment of length = (nР0# ± Р2)
By repeating Р3 times the line of fractal –Р2# and group No 4-3-2, we’ll get φ(Р2#) columns of groups No 4-3-2 of mod(Р2#) within the alternances ≤Р2 (Р3 line in the column).; By “eliminating” 1 number multiple–Р3 (in line of every column No 4–3-2), that is by transiting this group for mod(Р3#) with changing of its length from R(4,3) to R3,2 and alternances composition from ≤ Р2 to ≤ Р3. At the P.I. section from 1 to Р3# we’ll get the fractal –Р3#; of mod(Р3#). φ(Р3#) groups No 4-3-2:
Amount of groups of length and location	φ(Р3#) groups; No 4. R4. ≤ 2Р4 at every section <2Р5 is group No 4	φ(Р2#)*; *(Р3--3); R(3) ≤ 2Р2	+	2φ(Р2#); R(4) ≤ 2Р3	=	φ(Р3#); group No 3; R3 ≤ 2Р3	φ(Р2#)*; *(Р3--2); R(2) ≤ 2Р1	+	φ(Р2#); R(3) ≤ 2Р2	=	φ(Р3#); group No 2; R2 ≤ 2Р2
		At every section <2Р4 is group No 3					At every section <2Р3 is group No 2
Formula = n; Type max R; Groups and location	(Р3# ± Р4); max R4 = 2Р4; At the section with length = (Р3# ± Р5)	n = (multiple Р3 ± 1) / Р2# = the whole; (nР2#. ± .Р3) and (Р3--n)Р2#. ± .Р3; 2 groups No 3- max R3 = 2Р3.; On the segment of length = (nР2# ± Р4)					n = (multiple Р2*Р3 ± 1)/Р1# = the whole; (nР1# ± Р2) and (Р2Р3--n)Р1# ± Р2; 2 groups No 2- max R2 = 2Р2.; On the segment of length = (nР1# ± Р3)
…	…	…					…
And so on for the increasing meanings of modulus = mod Рn#. with: Рn# - primorial.; Р(…) < Р0 < Р1 < Р2 … < Рn are the consequent primes. С1-2-3-4 are primes and residues of mod(Рn#) R4-3-2 = (С4–3-2 – С1) that is length of the group = (R4-3-2--2)/2 of odd numbers.

Table 14.

The loopback of prime’s groups and residues rearrangement according to the increasing modulus. With max R = const from mod(Р₁#) to mod(Р₃#).

for the fractal −7# = 210, mod(7#), maxR3 = 2*7, at the P.I. section (СВ–СА) = 2*11, n = 3
СА = 79; АС = 109	…	С1 = 83.; 1С = 113	5,3	С2 = 89 и (7*13); (7*17) и 2С = 121	3,5	С3 = 97; 3С = 127	..	СВ = 101; ВС = 131
for the fractal −11# = 2310, mod(11#), maxR3 = 2*11, at the P.I. section (СВ–СА) = 2*13, n = 1
.СА = 197.; =n7#--13; .АС = 2087.; =11#--СВ	…	..С1 = 199.; =n7#--11; 1С = 2089.; =11#--С3	3,7,; 5,3	(11*19),С2 = 211; =n7#. ± .1; 2С = 2099,(11*191); =(11--n)7# ± 1	3,5,; 7,3,	..С3 = 221…; =n7# + 11; …3С = 2111; (11--n)7# + 11	..	.СВ = 223; =n7# + 13; ВС = 2113.; (11--n)7# + 13
for the fractal −13# = 30,030, mod(13#), maxR3 = 2*13, at the P.I. section (СВ–СА) = 2*17, n = 3
.СА = 6913.; =n11#--17; .АС = 23,083.; =13#--СВ	.; …	..С1 = 6917.; =n11#--13; 1С = 23,087.; =13#--С3	11,; 3,7,; 5,3	(13*533),С2 = 6931; =n11#. ± .1; 2С = 23,099,(13*1777); =(13--n)11# ± 1	3,5,; 7,3,; 11,	С3 = 6943; =n11# + 13; …3С = 23,113; (13--n)11# + 13	..	.СВ = 6947; =n11# + 17; ВС = 23,117.; (13--n)11# + 17
for the fractal −17# = 510,510,mod(17#), maxR3 = 2*17, at the P.I. section (СВ–СА) = 2*19,n = 2
.СА = 60,041.; =n13#--19; .АС = 450,431.; =17#--СВ	.; …	..С1 = 60,043.; =n13#--17; 1С = 450,433.; =17#--С3	13,; 11,; 3,7,; 5,3	С2 = 60,059,(17*3533); =n13#. ± .1; (17*26497),2С = 450,451; =(17--n)13# ± 1	3,5,; 7,3,; 11,; 13,	С3 = 60,077..; =n13# + 17; …3С = 450,467 (17-n)13# + 17	..	.СВ = 60,079; =n13# + 19; ВС = 450,469; (17--n)13# + 19
for the fractal −19# = 9,699,690,mod(19#), maxR3 = 2*19, at the P.I. section (СВ --СА) = 2*23, n = 1
СА = 510,487; АС = 9,189,157	…	С1 = 510,491; 1С = 9,189,161	17..; 5,3	С2 = 510,509 и кр.19.; Кр.19 и 2С = 9,189,181	3,5,; …17,	..С3 = 510,529 …3С = 9,189,199	..	.СВ = 510,533; ВС = 9,189,203

Table 15.

The numerical examples of the two line-symmetrical maximally long subgroups No 3 (containing 3 residues = С_1-2-3), according to the increasing modulus.

7. The maximal length of the P.I. section, where two maximally long subgroups No 2 (with 2 residues) for the mod(Р₃#)

Representing as one line, the first Р₂ lines in Table 1 we’ll get the fractal Р₂# according to the mod(Р₂#) – I.R.S. at mod(Р₂#), that is situated at the P.I. section from 1 to Р₂# and represented in 1 line of Table 2 where φ(Р₂#) of groups No 3 of mod(Р₂#) are situated with length R₃ ≤ 2Р₂ of the whole numbers. In this number, the two maximally long groups No 3 of mod(Р₂#) with length max R₃ = 2Р₂ of the whole numbers are represented. Then on the P.I. section from 1 to Р₃# (at Р₃ lines of Table 2), we’ll get φ(Р₂#) columns of group No 3 of mod(Р₂#), with length R₃ ≤ 2Р₂, (Р₃ lines in columns of groups No 3).

It is quite obvious, that in Section 9.1, it is proved that if by number Р₃, “eliminate,” that is to change for the model mod(Р₃#) residue С₂ in the column of every φ(Р₂#) group No 3 of mod(Р₂#), then at the P.I. section from 1 to Р₃#, that is at the fractal Р₃#, we’ll get φ(Р₂#) groups No 2 of mod(Р₃#) of the same length, that is R₃ ≤ 2Р₂ mod(Р₂#), would become R₂ ≤ 2Р₂ of mod(Р₃#) with changing the structure of alternance from ≤Р₂ to ≤Р₃.

As any eliminated residue С₂, during the rearrangement of groups from No 3 to No 2 for the mod(Р₃#) cannot change the length on no subgroup that is all R₂ would permanently ≤2Р₂, and included in φ(Р₂#) groups No 3 of mod(Р₂#) there are two uncial, repeated Р₃ times maximally long subgroups No 3 of mod(Р₂#) with the alternance ≤Р₂ with length maximal R₃ = 2Р₂, that would be rearranged into two maximally long line-symmetrical groups No 2 of mod(Р₃#) by “eliminating” the residues С₂ with the number multiple Р₃, (1 time in Р₃ lines). As all the other φ(Р₂#)--2 subgroups, rearranged from No 3 to No 2 for the mod(Р₃#), are shorter than (Р₂–1) of the off numbers, that is: R₂ < 2Р₂, 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(Р₂#) with length R₃ > 2Р₂. 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.

The length of these two line-symmetrical subgroups No 2 of mod(Р₃#), the length of alternance ≤Р₃, from С₁ to С₂, that is max R₂ = (С₂--С₁) = (nР₁# + Р₂)--(nР₁#--Р₂) =2Р₂; (Р₂–1) of odd numbers. Two of these subgroups No 2 are situated within the P.I. section (nР₁#. ± Р₃) with length (С_В–С_А) = 2Р₃; (Р₃–1) of odd numbers from С_А to С_В. The numerical values of these maximal groups No 2 is defined according to the formula: (multiple Р₂ and multiple Р₃) is (nР₁#. ± .1) and (Р₂Р₃--n)Р₁#. ± .1., with n and (Р₂Р₃–n) define the line number for the group No 2 of mod(Р₃#) in the column -Р₂*Р₃ of the duplication of the max group No 4 of mod(Р₁#) with the period = Р₁# (Table 1). That is n = (multiple Р₂*Р₃. ± 1)/Р₁# = the whole < Р₂*Р₃/2.

Herewith, it is quite obvious, and proved in Section 9.3, that all the other subgroups No 2 of mod(Р₃#), with different lengths, rearranges from groups No 3 would be within the P.I., with length not exceeding the limit =2Р₃ 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).

for the fractal - 7# = 210, mod(7#), maxR2 = 2*5, section (СВ–СА) = 2*7, n = 1
СА = --1; АС = 197	…	С1 = 1.; 1С = 199	3.	(5) и (7); (7*29) и (5*41)	3.	С2 = 11; 2С = 209	..	.СВ = 13; ВС = 211
for the fractal - 11# = 2310, mod(11#), maxR2 = 2*7, section (СВ–СА) = 2*11, n = 4
.СА = 109.; =n5#--11; .АС = 2179.; =11#--СВ	.; …	..С1 = 113.; =n5#--7; 1С = 2183.; =11#--С2	5,; 3.	(7*17),(11*11); =n5#. ± .1 = 120 ± 1; (11*191),(7*313); =(7*11--n)5# ± 1	3,; 5.	..С2 = 127…; =n5# + 7; …2С = 2197…; (7*11--n)5# + 7	..	.СВ = 131; =n5# + 11; ВС = 2201.; (7*11--n)5# + 11
for the fractal - 13# = 30,030, mod(13#), maxR2 = 2*11, section (СВ–СА) = 2*13, n = 45
.СА = 9437.; =n7#--13; .АС = 20,567.; =13#--СВ	.; …	..С1 = 9439.; =n7#--11; 1С = 20,569.; =13#--С2	3,; 7,; 5.; 3.	(11*859),(13*727); =n7#. ± .1 = 9450 ± 1; (13*1583),(11*1871); =(11*13--n)7# ± 1	3,; 5,; 7,; 3.	…С2 = 9461; =n7# + 11; …2С = 20,591..; (11*13-n)7# + 11	..	.СВ = 9463; =n7# + 13; ВС = 20,593..; (11*13--n)7# + 13
for the fractal - 17# = 510,510,mod(17#), maxR2 = 2*13, section (СВ–СА) = 2*17,n = 94
.СА = 217,123.; =n11#--17; .АС = 293,353.; =17#--СВ	.; …	..С1 = 217,127.; =n11#--13; 1С = 293,357.; =17#--С2	11,; 3,; 7,; 5,; 3.	(13*16703),(17*12773); =n11#. ± .1 = 217,140± 1; (17*17257),(13*22567); =(13*17--n)11# ± 1	3; 5,; 7,; 3,; 11,	..С2 = 217,153..; =n11# + 13; …2С = 293,383 (13*17--n)11# + 13	..	.СВ = 217,157; =n11# + 17; ВС = 293,387; (13*17--n)11# + 17
for the fractal - 19# = 9,699,690,mod(19#), maxR2 = 2*17, section (СВ --СА) = 2*19, n = 2
СА = 60,041; АС = 9,639,611	…	С1 = 60,043; 1С = 9,639,613	13; 3	кр.19., кр.17 = 60,061; Кр.17.,кр.19 = 9,639,631	3,; .13.	..С2 = 60,077; 2С = 9,639,647	..	.СВ = 60,079; ВС = 9,639,649

Table 16.

The numerical examples of the two line-symmetrical maximally long subgroups No 2 (containing 2 residues), according to the increasing modulus.

8. The loopback of rearrangement for φ(Р_n#) groups No 3 (with 3 residues) from mod(Р_n-1#) for mod(Р_n#)

The loopback order of rearrangement of φ(Р_n#) groups No 3, according to the increasing modulus, that are represented in column No 3 of Table 14 at Section 8 are examined by steps, for every recurrent increasing fractal -Р_n#:

During the transition from mod(Р₁#) to mod(Р₂#) of the fractal -Р₁# (1 line of Table 1) and every from φ(Р₁#) groups No 4-3-2 of mod(Р₁#) are repeated Р₂ times. That at the P.I. section from 1 to Р₂#, we’ll get φ(Р₁#) columns of No 4-3-2 groups (Р₂ limes at the column). Number Р₂ according to diagonals of Р₂ lines “eliminates,” that is rearranges to the mod(Р₂#) one time every of Р₂ repeated numbers of the P.I. section from 1 to Р₁# (1 number at every Р₂ line of every column No 4 and No 3), “eliminating” the residues С₁-С₂-С₃-С₄ in the groups No 4 (consult Section 8.1), and ALL numbers, besides the residues С₁-С₂-С₃ in the groups No 3 (consult Section 8.3).

8.1

It is quite obvious, that after “elimination” from every φ(Р₁#) column of the group No 4 of mod(Р₁#) one time every residue С₂ and С₃, we’ll get at Р₂ lines of every column of the No 4 groups of mod(Р₁#), –TWO groups No 3 of mod(Р₂#)

That is, we’ll get 2φ(Р₁#) subgroups No 3 of mod(Р₂#) with invariance length of the previous groups, that is R₄ φ(Р₁#) groups of No 4 mod(Р₁#), would become = R₃ for 2φ(Р₁#) groups No 3 of mod(Р₂#), with changing the alternance structure from ≤Р₁ to ≤Р₂, that are situated at the P.I section from 1 to Р₂# that is at the fractal-Р₂#.

Herewith within the 2φ(Р₁#) groups No 3 of mod(Р₂#), there are accounted all residues = С₁ and = С₄ of mod(Р₁#) and alternances ≤Р₂ of such rearranged groups from No 4 of mod(Р₁#) to No 3 of mod(Р₂#). As along of Р₂ duplication of the three adjoined groups No 4 of mod(Р₁#), represented in Table 3, we’ll get Table 17, with every residue = С₂ or = С₃, situated at one of the 3 lines of Table 3 (for example, at line а of Table 17), is accounted as residue С₁ or С₄ at the other two adjoined groups No 4 (at lines: b or с of Table 17), where they are situated within 4 consequent residues, going as the second or the third, that is they are “excluded” as С_{2 or 3} in these adjoined groups (lines):

• for line (b) С₂, С₃. (С₄ = multiple to Р₂), С₃ we’ll get the alternance ≤Р₂. R₃ < 2Р₂.

• for line (с) С₂, (С₁ = multiple to Р₂), С₂, С₃ we’ll get the alternance ≤Р₂. R₃ < 2Р₂.

Type-(a) group from 4 to No 3 (С1--С4)	СА	С1 (С2 = mtp.Р2) или (С3 = mtp.Р2), С4	СВ	with R4 is R3 (С4--С1) < 2Р2	at the P.I. section (СВ–СА) < 2Р3
Type-(b) group from 4 to No 3 (С2--С3)	С1	С2, С3, (С4 = multiple to Р2),С3	С2	with R4 is R3 (С3--С2) < 2Р2	at the P.I. section (С2–С1) < 2Р3
Type-(c) group from; 4 to No 3 (С2--С3)	С3	С2, (С1 = multiple to Р2) С2,С3	С4	with R4 is R3 (С3--С2) < 2Р2	at the P.I. section; (С4 – С3) < 2Р3

Table 17.

The representation of rearrangement of every 3 adjoined subgroups No 4 of mod(Р₁#) (represented in Table 3), while P₂ duplication of fractal -Р₁# from No 4 groups of mod(Р₁#) to No 3 groups of mod(Р₂#).

Thus, “eliminating” that means transition to mod(Р₂#), one time every 4 residue in φ(Р₁#) groups No 4 of mod(Р₁#), at the P.I. sections from 1 to Р₂#, that is at the fractal -Р₂#, we’ll, get the loopback, represented as 2φ(Р₁#) groups No 3 of mod (Р₂#) type: С_{1 рр}С_{2 рр}(С_2–3 = multiple Р₂)_ррС₃, with the alternances ≤Р₂, length R₃ ≤ 2Р₂. Including, pure 2 subgroups No 3 of mod(Р₂#) with length maximal R₃ = 2Р_2.

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(Р₂#): type (а) =(С_В–С_А) < 2Р₃; type (b) =(С₂–С₁) < 2Р₃; type (с) =(С₄–С₃) < 2₃.

Including pure two subgroups No 3 max R₃ = 2Р₂, located within the maximally long section with length = 2Р₃ of the whole numbers. (The rearrangement is studied in Section 6).

8.3

Within the Р₂ duplications φ(Р₁#) of the groups No 3 of mod(Р₁#), number Р₂ “eliminated,” that is rearranges to the mod(Р₂#) 1 time every of all previously eliminated numbers of group No 3, except three residues С₁--С₂–С₃. That is, transits to the mod(Р₂#) (Р₂--3) of No 3 groups in every φ(Р₁#) column of No 3 groups.

Then at the P.I. section from 1 to Р₂#, (that is included into fractal -Р₂#), we’ll get the loop back, represented as (Р₂--3) φ(Р₁#) of No 3 groups repetition for the mod(Р₂#) with “changing” the alternance from ≤Р₁ to ≤Р₂. Without changes the groups No 3 length for the mod(Р₂#), R₃ = const and numbers composition within the alternances ≤Р₂; that is the previously “eliminated” ≤ Р₁, according to the 1 least >1 from the number are accounted. Type: С_{1 рр}С_{2 рр}С₃.

With: R₃ = const =? (with to mod(Р₂#) R₃ < < R₄ = 2Р₂).

8.4

Total at the fractal -Р₂# at the P.I. section from 1 to Р₂# we’ll get the loopback of the rearranged groups No 3 for the mod(Р₂#) represented in Sections 8.1 and 8.3.

That is: 2φ(Р₁#) (Section 8.1 with the length = R₃ ≤ 2Р₂) + φ(Р₁#)(Р₂–3) (Section 8.3 with the length R₃ < < R₄ = 2Р₂) = Р₂φ(Р₁#)--3φ(Р₁#) + 2φ(Р₁#) = Р₂ φ(Р₁#)--φ(Р₁#) = φ(Р₁#)(Р₂–1) = φ(Р₂#) of the subgroups No 3 for the mod(Р₂#), included at the alternances ≤Р₂ with length R₃ ≤ 2Р₂, including two maximally long subgroups No 3 max R₃ = 2Р₂.

8.5

As all φ(Р₂#) of the subgroups No 3 of mod(Р₂#) are examined in Sections 8.1 and 8.3, with all eliminated one time residues С₂ or С₃, examined in Section 8.1, cannot change the length = R₃ ≤ 2Р₂, none of the 2φ(Р₁#) groups, rearranged from No 4 to No 3 for the mod(Р₂#). Herewith the residues С₁ and С₄ are also accounted in two adjoining groups of Table 17 as С₂ or С₃. And indiscriminately φ(Р₁#)(Р₂–3) groups No 3 for the mod(Р₂#), examined in Section 8.3 are shorter than limit R₃ = 2Р₂.

So, there are no other ways to make groups No 3 of mod(Р₂#) with length R₃ > 2Р₂, besides the way to form the maximally long subgroups No 3 with length max R₃ = 2Р₂, represented in Sections 6 and 8.1.

8.6

Thus we got, that for every recurrent = Р₂, φ(Р₂#) residues of mod(Р₂#) situated in the fractal -Р₂# (Р₂ lines of Table 1), represented as loop back φ(Р₂#) of subgroups No 3 (3 residues of mod(Р₂#), represented in Section 8.4), that is pure THREE consequent prime (_A,B,CС) type: P₂ < (_AС-_BС-_CС) < P₃² at the P.I. section from Р₂ to Р₃². And further, from Р₃² to Р₂#, pure THREE consequent residue of mod(Р₂#) at every P.I. section, with length not exceeding 2Р₃ of the whole numbers (see Section 8.2), with length of every subgroup No 3 at every section is R₃ = (_CС--_AС) ≤ 2Р₂ (consult Sections 8.1 and 8.4).

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).

9. 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 Р₂ lines in Table 1 as one line, we’ll get the fractal -Р₂# according to mod(Р₂#)-I.R.S of mod(Р₂#) at the P.I. sections from 1 to Р₂# (1 line of Table 2).

With φ(Р₂#) line-symmetrical the least residues of mod(Р₂#), which according to Section 2 are indexed according to φ(Р₂#) groups of residues No 4-3-2 for the mod(Р₂#).

At the transition from mod(Р₂#) to mod(Р₃#), the fractal -Р₂# and every from the φ(Р₂#) groups No 4-3-2 mod(Р₂#) are repeated Р₃ times. Then at the P.I. section from 1 to Р₃# we’ll get φ(Р₂#) columns of No 4-3-2 groups (Р₃ lines at the column). Number Р₃ according to diagonals Р₃ lines “eliminates,” that is transits for the mod (Р₃#) one time every Р₃ of the duplicated numbers of the P.I. section from 1 to Р₂#. (One number at every Р₃ line of every column No 3 and No 2), “eliminating” the residues С₁-С₂-С₃ at the groups No 3 (consult Section 9.1) and ALL numbers, except the residues С₁-С₂ at the groups No 2 (consult Section 9.3).

9.1

It is quite obvious, that after “elimination” in every φ(Р₂#) column of group No 3 of mod(Р₂#), 1 time the residue - С₂, we’ll get in Р₃ line of every column of groups No 3 of mod(Р₂#), one group No 2 of mod(Р₃#), that is totally we’ll get φ(Р₂#) subgroups No 2 of mod(Р₃#) with invariance length of previous groups, that is R₃ φ(Р₂#) groups No 3 of mod(Р₂#) would become = R₂ for φ(Р₂#) groups No 2 of mod(Р₃#) with changing the alternance composition from ≤Р₂ to ≤Р₃, that are situated at P.I. section from 1 to Р₃# that is at the fractal -Р₃#

Herewith in φ(Р₂#) groups No 2 of mod(Р₃#) are accounted all residues = С₁ and = С₃ of mod(Р₂#) and alternances ≤Р₃ of such “rearranged” groups from No 3 of mod(Р₂#) to No 2 of mod(Р₃#). As in the course of Р₃ duplication of two adjoined groups No 3 of mod(Р₂#), represented in Table 5, we’ll get the table No 11 with each of the residues = С₂, situated on one of two lines of Table 5 (for example, the line (а) Table 18, is accounted as the residue С₁ or С₃ at the other adjoined group No 3 (in line b). of Table 12), where it is represented in 3 consequent residues as the second one, that is “excluded” as С₂ at this adjoined group (line):

Type-(a) group from 3 to No 2 (С1--С3)	СА	С1, (С2 = multiple Р3)., С3	СВ	with R3 is R2 (С3--С1) < 2Р2	at the P.I. section (СВ–СА) < 2Р3
Type (a) group from 3 to No 2 (2С–С2)	1С	2С.,(3С1 = multiple Р3).,С2	С1	with R3 is R2 (С2--2С) < 2Р2	At the P.I. section (С1–1С) < 2Р3

Table 18.

Representation of rearrangement of any 2 adjoined subgroups No 3 of mod(Р₂#) (represented in Table 5), within P₃ duplications of fractal -Р₂#. From groups No 3 of mod (Р₂#) to the groups No 2 of mod (Р₃#).

-for line (b) ₂С., (₃С = multiple Р₃)., С₂, we’ll get the alternance ≤Р₃. R₂ < 2Р₂

Thus, after “elimination” that is transferring to mod(Р₃#), one time for every of 3 residues at φ(Р₂#) groups No 3 of mod(Р₂#), at the P.I. sections from 1 to Р₃#, that is in fractal -Р₃#, we’ll get the loop back in the form of φ(Р₂#) groups No 2 of mod(Р₃#), type: С_{1 рр}(С₂ = multiple Р₂)_ррС₃, with the alternances ≤Р₃, with length R₂ ≤ 2Р₂. Including pure 2 subgroups No 2 of mod(Р₃#) with length maximal R₂ = 2Р_2.

9.2

Herewith it is quite obvious that any two consequent residues of any subgroup No 2 according to the increasing mod(Р_n#), represented in Table 18 are still within the P.I. section with length not exceeding - 2Р_n + 1 of the whole numbers, as the “eliminated” residues С₂ and С_1;3 of mod(Р_n#) doesn’t change the location of any subgroup No 2. That is, in two adjoined groups No 2 lines of Table 18 for mod(Р₃#) we get: type (а) =(С_В–С_А) < 2Р₃; type (b) =(С₁–₁С) < 2Р₃.

Including pure two subgroups No 2 max R₂ = 2Р₂, located within the maximally long section with length = 2Р₃ of the whole numbers, two rearrangement is studies in Section 7.

9.3

Along with Р₃ duplications φ(Р₂#) of groups No 2 of mod(Р₂#), the number Р₃ “eliminates” that is transits to the mod(Р₃#) 1 time every of all previously eliminated numbers of every group No 2, except two residues С₁--С₂. That it, it transits to mod(Р₃#) (Р₃--2) of groups No 2 in every φ(Р₂#) column of No 2 groups.

Then at the P.I. section from 1 to Р₃# that is within the fractal -Р₃# we’ll get the loopback, represented as (Р₃--2)φ(Р₂#) duplications of No 2 groups for the mod (Р₃#) with the alternance “changes” from ≤Р₂ to ≤Р₃. Without changing the length of groups No 2 for mod(Р₃#), R₂ = const and numbers composition at the alternances ≤Р₃ (as previously eliminated ≤Р₂, for the 1 least >1 from the number is accounted). Type: С_{1 рррр}С₂. With: R₂ = const =? (with to mod(Р₃#) R₂ < < R₃ = 2Р₂).

9.4

Totally at the fractal Р₃# at P.I. section from 1 to Р₃# we’ll get the loopback of the rearranged groups No 2 for mod(Р₃#) represented in Sections 9.1 and 9.3. That is: φ(Р₂#) (Section 9.1 with length = R₂ ≤ 2Р₂) + φ(Р₂#)(Р₃–2) (Section 9.3 with length R₂ < <R₃ = 2Р₂) = Р₃φ(Р₂#)--2φ(Р₂#) + φ(Р₂#) = Р₃ φ(Р₂#)--φ(Р₂#) = φ(Р₂#)(Р₃–1) = φ(Р₃#) of the subgroups No 2 for mod(Р₃#), located within the alternances ≤Р₃ with length R₂ ≤ 2Р₂, including two maximally long subgroups No 2 max R₂ = 2Р₂.

9.5

As all φ(Р₃#) of the subgroups No 2 of mod(Р₃#) are examines in Sections 9.1 and 9.3, with every eliminated one time in the column residue С₂ examined in Section 9.1, can change the length = R₂ ≤ 2Р₂, of none of φ(Р₂#) groups, rearranged from No 3 to No 2 for mod(Р₃#), herewith residues С₁ and С₃ are excluded at the adjoined group of Table 18 as С₂ and indiscriminately (Р₃–2)φ(Р₂#) groups No 2 of mod(Р₂#), examined in Section 9.3 are shorter than limit R = 2Р₂.

Thus, there are no other ways of making groups No 2 of mod(Р₃#) with length R₂ > 2Р₂, but constructing two maximally long subgroups No 2 with length max R₂ = 2Р₂, as examined in Sections 7 and 9.1.

9.6

And so we get, that for every recurrent prime = Р₃, φ(Р₃#) residues of mod (Р₃#) are in the fractal -Р₃# (in Р₃ lines of Table 2), represented as loopback φ(Р₃#) of the subgroups No 2 (2 residues of mod(Р₃#) are indicated in Section 9.4). Thus, pure TWO consequent primes (_A,BС) of type: P₁, P₂ < (_AС-_BС) < P₃² at the P.I. section from Р₂ to Р₃², and further, from Р₃² to Р₃# pure TWO consequent residues of mod (Р₃#), at every P.I. sections with length not exceeding 2Р₃ of the whole numbers (consult Section 9.2), with length of every subgroup No 2 at every section is: R₂ = (_ВС--_AС) ≤ 2Р₂ (consult Sections 9.1 and 9.4).

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).

10. Proof of section (b) and section (с) of Theorem 1

While examining the P.I., represented in the form of alternance (array) of primes (according to the 1 least prime factor > 1 from every whole number), we’ll get that for every recurrent prime -Р_n; the P.I. is the line-symmetrical primary-repeated fractal -Р_n#, located at the P.I. section from 1 to Р_n#, represented as φ(Р_n#) of the I.R.S. residue of mod(Р_n#), between which the P.I. sections are situated (with different length), represented as the alternances (array) of different amounts of different first primes ≤Р_n – NOT residues mod(Р_n#).

By indexing φ(Р_n#) of the least residues of every recurrent fractal -Р_n#, groups pure with 4, 3, and 2 consequent residue of mod(Р_n#) (analogously as in Section 2). We’ll get that every recurrent fractal -Р_n# has got three types of arrays of the subgroups of residues of mod(Р_n#): with φ(Р_n#) groups: No 4 (with 4 residues), No 3 (with 3 residues), and No 2 (with 2 residues, repeated without changes with the period = Р_n#).

At every recurrent transition, for example, from mod(Р₂#) to mod(Р₃#), the first line of fractal -Р₂# and group No 4-3-2 of mod(Р₂#) are repeated Р₃ times (consult Р₃ lines of Table 2). At the P.I. section from 1 to Р₃# we’ll get φ(Р₂#) columns of groups No 4-3-2 of mod(Р₂#) within the alternances ≤Р₂ (Р₃ lines in column), “eliminating” 1 number multiple to -Р₃ (in line of every column No 4-3-2), that is by transition of this group to mod(Р₃#) with changing of its length: from R_(4,3) to R_3,2 and the alternance composition from ≤Р₂ to ≤Р₃, we’ll get φ(Р₃#) groups No 4-3-2 of mod(Р₃#).

The rearrangement order of these subgroups No 4-3-2 at the increasing modulus is proved in Sections 5, 6, 7, 8, and 9. Representing as one line of the P.I. section from 1 to Р₃# we’ll get the fractal -Р₃# of mod(Р₃#), with φ(Р₃#) groups No 4-3-2 of mod(Р₃#).

And so on for every recurrent prime = Р_n, the results of proof are demonstrated in Sections from 5 to 9 and for visualization they are grouped together in Table 14.

As far as we know that for every recurrent prime -Р_n = Р₍₁₎, the P.I. is the line-symmetrical primordial repeated fractal -Р₍₁₎#, located at the P.I. section from 1 to Р₍₁₎#. There are φ(Р₍₁₎#) residues of mod(Р₍₁₎#) between which are located the alternances (arrays) of primes ≤ Р₍₁₎ (1 the least>1 from every NOT residue of mod (Р₍₁₎#)).

According to Section 5, there is the only maximally long subgroup No 4 (with 4 residues) of mod(Р₍₁₎#) with length maxR₄ = 2Р₍₂₎.

Let us assume that there are such primes Р₍₂₎ or Р₍₃₎, for which at the P.I., represented as alternance ≤ Р₍₂₎, in the form of fractal –Р₍₂₎# we can form more than two maximally long subgroups No 3 of mod(Р₍₂₎#), with: (or) R₃ > 2Р₍₂₎ or at the fractal –Р₍₃₎# with P.I. is represented by alternances ≤ Р₍₃₎, we can compare more than maximally long subgroups No 2 of mod(Р₍₃₎#), with: (or) R₂ > 2Р₍₂₎.

Then, in the course of the opposite reduction of the modulus, that is at the result of Р₍₂₎*Р₍₃₎ repetition of such subgroups as No 3 or No 2 with the repetition period Р₍₁₎# (for the downward meanings of numbers) and backing up the Р₍₂₎ and Р₍₃₎ numbers as residues (according to the decreased moduli)^**, that are situated in Р₍₂₎*Р₍₃₎ lines analogues to Table 1. At the upper lines of such columns, consisting of Р₍₂₎*Р₍₃₎ lines, we’ll get the P.I as fractal-Р₍₁₎#, represented by alternances ≤ Р₍₁₎, where at the P.I. section from 1 to Р₍₁₎# would be located more than one subgroup No 4 (with 4 residues) of mod(Р₍₁₎#) or subgroups No 4, with R₄ > 2Р₍₂₎. It is quite obvious, that any such group No 4 according to the reestablished mod(Р₍₁₎#) would be line-symmetrical to the left and to the right from the symmetry center of number = Р₍₁₎#/2, that means formed by two different ways, that contradicts to the axiom set.

^**Number Р₍₃₎ of the fractal -Р₍₃₎# -NOT residue of mod(Р₃#), located in the group No 2 of mod(Р₍₃₎#) within the alternance ≤ Р₍₃₎ with length R₍₂₎ > 2Р₍₂₎, and for mod(Р₍₂₎#) would be accounted as the third residue in the group No 3 within the alternance ≤Р₍₂₎, without changing the length of this group No 3 with R₍₂₎ is R₃ > 2Р₍₂₎.

Number Р₍₂₎ of the fractal -Р₍₂₎# -NOT residue of mod(Р₂#) is located in the group No 3 of mod(Р₍₂₎#) within the alternance ≤Р₍₂₎, with length R₃ > 2Р₍₂₎, and for mod

(Р₍₁₎#), would be accounted as the fourth residue in the group No 4 within the alternance ≤ Р₍₁₎, without changing the length of this group No 4 with R₍₃₎ is R₄ > 2Р₍₂₎.

11. Conclusion

Thus, Theorem 1 allowed us to prove the existence of a new law in mathematics – “on the plume distribution of Prime numbers.” Since the methods used in number theory do not allow us to approach the problem of the distribution of prime numbers, it means that further expansion of the method proposed in the article for studying the natural series of numbers will simplify and solve many other problems that are not solved in mathematics.

So from Theorem 1 “Loopback of primes distribution” follows:

Theorem No 2. For every whole number = N at the P.I. section from 1 to N + $2\sqrt{N}$:

1. Primes are located as groups, pure three consequent primes of (Р₁-Р₂-Р₃) type. Herewith the distance from the first to the third prime of every group is less than $2\sqrt{N}$ of the whole numbers, that is (Р₃–Р₁) < $2\sqrt{N}$ whole numbers.

2. The same primes are distributed as the loopback, pure two consequent primes at every P.I. sections, shorter than $2\sqrt{N}$ whole numbers.

Proof. Every whole number = N is located within the squared two consequent primes: Р₁² < N ≤ Р₂² with: $2\sqrt{N}$ >2Р₁.

That means every N is located within the fractal -Р₁#. Then:

1. From the section (b) of the theorem “Loopback of primes distribution” follows, that at fractal -Р₁# of mod(Р₁#) at the P.I. section from 1 to Р₂² ≥ (N + $2\sqrt{N}$), at every P.I. section with length not exceeding 2Р₂ of the whole numbers is located at the subgroup from three consequent primes of (Р₁-Р₂-Р₃) type, with length of every subgroup, that is distance from the first to the third prime of every subgroup doesn’t exceed 2Р₁ whole numbers, that is (Р₃–Р₁) < 2Р₁ whole numbers. As length of every section $2\sqrt{N}$ > length of the section = 2Р₁. Then from 1 to N + $2\sqrt{N}$ – every:

   (Р₃–Р₁) < $2\sqrt{N}$ of the whole numbers.

2. From the section (с) of the Theorem “Loopback of primes distribution” it follows, that at the fractal -Р₁# of mod(Р₁#) at the P.I. section from 1 to Р₂² ≥ (N + $2\sqrt{N}$) at every I.P. section with length not exceeding 2Р₁ of the whole numbers, there is the subgroup from two consequent primes Р₁ and Р₂. As $2\sqrt{N}$ whole numbers >2Р₁ whole numbers, that means, that at the P.I. section from 1 to N + $2\sqrt{N}$ at every P.I. section with length not exceeding $2\sqrt{N}$ whole numbers, there is the loopback of primes, represented as the subgroup for two consequent primes.

Genuinely: Every P.I. section with length = 2$\sqrt{N}$ of the whole numbers is located at the fractal -Р₁# at the P.I. section. <Р₂² as, Р₁² < (N + 2$\sqrt{N})$ ≤ Р₂², with: $2\sqrt{N}$ > 2Р₁.

It is feasible, that there is a P.I. section with length = 2$\sqrt{N}$ 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$\sqrt{N}$ of the whole numbers >2Р₁, but this contradicts to section (с) of the Theorem “Loopback of primes distribution,” that states, that is every fractal -Р₁# according to mod(Р₁#), on every P.I. sections with length not exceeding 2Р₁ of the whole numbers, there is a subgroup No 2 with 2 residues of mod(Р₁#), that is two primes <Р₂².