Open access peer-reviewed chapter

# Prime Numbers Distribution Line

Submitted: April 14th 2020Reviewed: April 23rd 2020Published: September 29th 2020

DOI: 10.5772/intechopen.92639

## 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.

### Keywords

• residue groups
• prime numbers
• primorial
• sieve of Eratosthenes
• alternations
• fractal

## 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 = 13,5,7..С2 = Р2pрр..С3..ррр.. Сn ..рррР1#–СnрррР1#–С2..7,5.31#–1)
1 + Р1#3,5,7..С2 + Р1#pррС3 + Р1#рррСn + Р1#ррр2Р1#–Сnррр2Р1#–С2..7,5.3(2Р1#–1)
1 + 2Р1#3,5,7..С2 + 2Р1#pррС3 + 2Р1#рррСn + 2Р1#ррр3Р1#–Сnррр3Р1#–С2..7,5.3(3Р1#–1)
3,5,7..pррррррррррр..7,5.3
3,5,7..pррррррррР2#–СnрррР2#–С2..7,5.32#–1)
1 + Р2#3,5,7..С2 + Р2#pррС3 + Р2#рррСn + Р2#рррррр..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 = 13,5,7,,,С2 = Р3pрр..С3..ррр.. Сn ..рррР2#–СnрррР2#–С2..7,5.32#–1)
1 + Р2#3,5,7,,,С2 + Р2#pррС3 + Р2#рррСn + Р2#ррр2Р2#–Сnррр2Р2#–С2..7,5.3(2Р2#–1)
1 + 2Р2#3,5,7,,,С2 + 2Р2#pррС3 + 2Р2#рррСn + 2Р2#ррр3Р2#–Сnррр3Р2#–С2..7,5.3(3Р2#–1)
3,5,7,,,pррррррррррр..7,5.3
3,5,7,,,pррррррррР3#–СnрррР3#–С2..7,5.33#–1)
1 + Р3#3,5,7,,,С2 + Р3#pррС3 + Р3#рррСn + Р3#рррррр..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: Сn to the left from the number = Р1#/2 and (Р1#–Сn) to the right from the number = Р1#/2, with Сn – 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 = Рn has got its periodical fractal = Рn# 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(Рn#) of the least residues of mod(Рn#) type = С and (Рn#–С), of every recurrent fractal = Рn#, gradate P.I. as φ(Рn#) “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(Рn#), or un the form of alternance (array) of the first primes  Рn, (according to the 1st least prime factor >1 from every NOT residue of mod(Рn#)), hereinafter the alternance Рn. С – residue of mod(Рn#) (according to the 1st least >Рn from every residue of mod(Рn#)), location from 1 to Рn# is line symmetrical relating to number = Рn#/2. And further, repeated without rearrangement of their position with the period = Рn#. Then, after we define the overall - maximal length of alternance that we can form using the fist primes р  Рn, (NOT residues of mod (Рn#)), type С1…3рр3рр3рр3…С2 that is maximal amount of consequent odd numbers = maximal length of alternance -р  Рn (one least NOT residue of mod(Рn#) >1 from the number), we can evaluate the distance between every two consequent residues of mod (Рn#) that is between two primes <(Рn + 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(Рn#)).

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(Рn#) for every fractal Рn#.

In this scientific work, the φ(Рn#) of the least residue of mod(Рn#) of every recurrent fractal -Рn# 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(Рn#). These groups No 4-3-2 are analyzed as subgroups with No 4-3-2 consequent residues of mod(Рn#) that are surrounded by the maximal permissible amount of consequent NOT residues of mod(Рn#).

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 Рn, (that is maximal permissible amount of all NOT residues of mod(Рn#)), situated between two residues from СА to СВ, between which, as subgroups are situated the groups of residues of mod (Рn#). Type:

1. -No 4: СА..3рр3..Р1..Р2..Р3..Р4..3рр3..СВ.

2. No 3: СА..3рр3..Р1..Р2..Р3..3рр3..СВ.

3. No 2: СА..3 рр3..Р1..Р2..3рр3..СВ.

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… P22 1С.3рр3.4С.3рр3.. Р1# 3,5,7… -------BP..3рр3..CP..3рр3..DP..3рр3..AP… P22 ---2С..3рр3..5С..2C Р1# 5,7… - -----CP..3рр3..DP.3рр3..AP..3рр3..BP P22 -----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: ..3рр3..BС..3рр3.. .. 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: ..3рр3..BС..3рр3... 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: ..3рр3... 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 79).

 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....3рр3..BP ..3рр3..CPP221С..3рр3..3С…3рр3.. 1С…3СР1#
3.,5.,7…--------.BP ..3рр3..CP..3рр3..АP..3рр3..P22--------2С..3рр3..4С.. 2С .. 4СР1#
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....3рр3..BP..3рр3..АP.3рр3..P221С..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....3рр3..BP..3рр3..АP.3рр3..P321С..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....3рр3..BP..3рр3..АP.3рр3..P421С..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#): СА – to the left and СВ – 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 СА to СВ: (а)СА-(С1234)-СВ. (b) СА-(С123)-СВ. (с) СА-(С12)-СВ.

## 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 -Рn#), of type is defined:

max R4 = (С4 – С1) = 2Рn + 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 РnFractal
- 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)
Р1Р1#mod(Р1#).max R4 =
4–С1) = 2Р2
>>max R3 =
3–С1) =?
>>max R2 =
2–С1) =?
Р2Р2#mod(Р2#).max R4 =
4–С1) = 2Р3
>>max R3 =
3–С1) =?
>>max R2 =
2–С1) =?
Р4Р4#mod(Р4#).max R4 =
4–С1) = 2Р5
>>max R3 =
3–С1) =?
>>max R2 =
2–С1) =?
Р5Р5#mod(Р5#).max R4 =
4–С1) = 2Р6
>>max R3 =
3–С1) =?
>>max R2 =
2–С1) =?
>>>>
РnРn#mod(Рn#).max R4 =
4–С1) = 2Рn + 1
>>max R3 =
3–С1) =?
>>max R2=
2–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 -Рn# 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:

R4=(DС–AС)2Р2.

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

• 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.

• 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. R4-3-2 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 (R4,3,2–2)/2 odd numbers.

The order of groups (а), (b), (с) rearrangement according to the increasing modulus for visual clarity is indicated in Table 11.

12345
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 + 12)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 + 12φ(Рn#) groups No 3 for 3 residues of mod(Рn#) (containing 3 simple) AС-BС-CС < Pn + 12φ(Рn#) groups No 2 for 2 residues of mod(Рn#) (containing 2 simple) (AС-BС) < Pn + 12
Subgroup No 4 length. R4 = DС--AСLength of every section for the group No 4Subgroup No 3 length. R3 = CС--AСLength of every section for the group No 3Subgroup No 2 length. R2 = BС--AСLength of every section for the group No 2
1 ÷ Рn# mod(Рn#)from (Рn)2 to (Рn + 1)2≥4Рn + 1 numberR4 ≤ 2Pn + 1
<(Pn± Pn + 1)
≤2Pn + 2 <(Pn± Pn + 2)R3 ≤ 2Pn
(Table 12)
≤2Pn + 1 numberR2 ≤ 2Pn–1 (Table 13)≤2Pn number
1 ÷ Р1# mod(Р1#)from (Р1)2 to (Р2)2≥4Р2 numberR4 ≤ 2P2≤2P3 numberR3 ≤ 2P1≤2P2 numberR2 ≤ 2P0≤2P1 number
1 ÷ Р2# mod(Р2#)from (Р2)2 to (Р3)2≥4Р3 numberR4 ≤ 2P3≤2P4 numberR3 ≤ 2P2≤2P3 numberR2 ≤ 2P1≤2P2 number
1 ÷ Р3# mod(Р3#)from (Р3)2 to (Р4) 2≥4Р4 numberR4 ≤ 2P4≤2P5 numberR3 ≤ 2P3≤2P4 numberR2 ≤ 2P2≤2P3 number
1 ÷ Р4# mod(Р4#)from (Р4)2 to (Р5) 2≥4Р5 numberR4 ≤ 2P5≤2P6 numberR3 ≤ 2P4≤2P5 numberR2 ≤ 2P3≤2P4 number
1 ÷ Р5# mod(Р5#)from (Р5)2 to (Р6) 2≥4Р6 numberR4 ≤ 2P6≤2P7 numberR3 ≤ 2P5≤2P6 numberR2 ≤ 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 ≤ Рn (NOT residues of mod(Рn#)), 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(Рn#). That is that maximal amount of NOT residues of mod(Рn#), maximal long alternance Рn.

Then for every recurrent prime number Рn = 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(Рn#)), shall be situated within the P.I. section from СА to СВ, with as the subgroup is the only maximal long maximal subgroup No 4 (С1234) with the 4 consequent residue of mod(Р1#): Type: СА = (Р1#--Р3)..С1 = (Р1#--Р2).

…С2 = (Р1#--1),С3 = (Р1# + 1) … С4 = (Р1# + Р2)… СВ = (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 СА to СВ would be situated maximal as well as all the other φ(Р1#) subgroups No 4 (with 4 residues) of mod(Р1#), is: (СВ–СА) = (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 Рn 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 СА to СВ), 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#) (С1234). Type: СА = (Р1#-Р3)…3рр3… С1 = (Р1#-Р2)…3рр3…С2 = (Р1#-1), С3 = (Р1# + 1)…3рр3…С4 = (Р1# + Р2)… …3рр3…СВ = (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 СА to СВ with length not exceeding the maximal long P.I. section (СВ–СА) ≤2Р3 of the whole numbers, not exceeding (Р3–1) of the odd numbers.

And so on, for all posterior prime numbers = Рn, at the increasing fractals -Рn# 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 Рn = 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#--Р3АС = Р2#--СВ(Р2--n)Р1#--Р3 .. ……С1…=nР1#--Р21С = Р2#--С3(Р2--n)Р1#--Р2 11,3,7,5,3 (кр.Р2) и С2=nР1#. ± .12С и (кр.Р2)(Р2-n)Р1# ± 1 3,5,7,3,11. ……С3…=nР1# + Р23С = Р2#--С1(Р2--n)Р1# + Р2 .. ….СВ……=nР1# + Р3ВС = 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#--Р4АС = Р3#--СВ(Р3--n)Р2#--Р4 .. ……С1…=nР2#--Р31С = Р2#--С3(Р3--n)Р2#--Р3 11,3,7,5,3 (кр.Р3) и С2=nР2#. ± .12С и (кр.Р3)(Р3--n)Р2# ± 1 3,5,7,3,11, ……С3…=nР2# + Р33С = Р2#--С1(Р3--n)Р2# + Р3 .. ….СВ……=nР2# + Р4ВС = 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#--Р3АС = Р3#--СВ(Р2Р3--n)Р1#--Р3 .. ……С1…=nР1#--Р21С = 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# + Р22С = Р3#--С1(Р2Р3--n)Р1# + Р2 .. ….СВ……=nР1# + Р3ВС = 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#--Р4АС = Р4#--СВ(Р4Р3--n)Р2#--Р4 .. ……С1…=nР2#--Р31С = 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# + Р32С = Р4#--С1(Р4Р3--n)Р2# + Р3 .. ….СВ……=nР2# + Р4ВС = 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 (СВ–СА) = 2Р3; (Р3–1) of the odd numbers from СА to СВ. 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 = Рn, at the increasing fractals Рn#, with n is the whole, represented in Tables 11 and 14 (the proof is indicated in Section 10). (Numerical illustrations are in Table 15).

1234
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 3At 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 locationφ(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 3At 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 locationφ(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 3At 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
5,3С2 = 89 и (7*13)
(7*17) и 2С = 121
3,5С3 = 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
А = 197.
=n7#--13
С = 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
..В = 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
А = 6913.
=n11#--17
С = 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
..В = 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
А = 60,041.
=n13#--19
С = 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
..В = 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..В = 510,533
ВС = 9,189,203

### Table 15.

The numerical examples of the two line-symmetrical maximally long subgroups No 3 (containing 3 residues = С1-2-3), 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 (СВ–СА) = 2Р3; (Р3–1) of odd numbers from СА to СВ. 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 = Рn, at the increasing fractals Рn#, 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 = --1АС = 197 … С1 = 1.1С = 199 3. (5) и (7)(7*29) и (5*41) 3. С2 = 112С = 209 .. .СВ = 13ВС = 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#--71С = 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# + 11ВС = 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#--111С = 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# + 13ВС = 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#--131С = 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 35,7,3,11, ..С2 = 217,153..=n11# + 13…2С = 293,383(13*17--n)11# + 13 .. .СВ = 217,157=n11# + 17ВС = 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 = 60,041АС = 9,639,611 … С1 = 60,0431С = 9,639,613 133 кр.19., кр.17 = 60,061Кр.17.,кр.19 = 9,639,631 3,.13. ..С2 = 60,0772С = 9,639,647 .. .СВ = 60,079ВС = 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(Рn#) groups No 3 (with 3 residues) from mod(Рn-1#) for mod(Рn#)

The loopback order of rearrangement of φ(Рn#) 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 -Рn#:

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 С1234 in the groups No 4 (consult Section 8.1), and ALL numbers, besides the residues С123 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 С2 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)САС12 = mtp.Р2) или3 = mtp.Р2), С4СВwith R4 is R34--С1) < 2Р2at the P.I. section (СВ–СА< 2Р3
Type-(b) group from 4 to No 3 (С2--С3)С1С2, С3, (С4 = multiple to Р2),С3С2with R4 is R33--С2) < 2Р2at the P.I. section (С2–С1) < 2Р3
Type-(c) group from
4 to No 3 (С2--С3)
С3С2, (С1 = multiple to Р2) С23С4with R4 is R33--С2) < 2Р2at the P.I. section
4 – 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: С1 ррС2 рр2–3 = multiple Р2)ррС3, with the alternances ≤Р2, length R3 ≤ 2Р2. Including, pure 2 subgroups No 3 of mod(Р2#) with length maximal R3 = 2Р2.

8.2

Herewith, it is quite obvious that every three consequent residues of every subgroup No 3 according to the increasing group of mod(Рn#), represented in Table 17, are still within the P.I. section with length not exceeding - 2Рn + 1 of the whole numbers, as “eliminated” residues С2;3 and С1;4 of mod(Рn#) 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) =(СВ–СА< 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: С1 ррС2 ррС3.

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 = Рn, represented as the loopback of groups distribution of the residues No 3 at the increasing fractals -Рn# according to the increasing meanings of modulus-mod(Рn#) 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(Рn#) groups No 2 (2 residues) from the mod(Рn-1#) to mod(Рn#)

The looped back order of rearrangement φ(Рn#) 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 -Рn#:

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 С123 at the groups No 3 (consult Section 9.1) and ALL numbers, except the residues С12 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)САС1, (С2 = multiple Р3)., С3СВwith R3 is R23--С1) < 2Р2at the P.I. section (СВ–СА< 2Р3
Type (a) group from 3 to No 2 (2С–С2)1С2С.,(3С1 = multiple Р3).,С2С1with R3 is R22--2С) < 2Р2At the P.I. section (С11С) < 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: С1 рр2 = multiple Р2)ррС3, with the alternances ≤Р3, with length R2 ≤ 2Р2. Including pure 2 subgroups No 2 of mod(Р3#) with length maximal R2 = 2Р2.

9.2

Herewith it is quite obvious that any two consequent residues of any subgroup No 2 according to the increasing mod(Рn#), represented in Table 18 are still within the P.I. section with length not exceeding - 2Рn + 1 of the whole numbers, as the “eliminated” residues С2 and С1;3 of mod(Рn#) 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) =(СВ–СА< 2Р3; type (b) =(С11С) < 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: С1 ррррС2. 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 = Рn, in the form of loopback of residues of groups No 2 distribution at the increasing fractals -Рn#, with the increasing values of modulus of mod(Рn#), 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 -Рn; the P.I. is the line-symmetrical primary-repeated fractal -Рn#, located at the P.I. section from 1 to Рn#, represented as φ(Рn#) of the I.R.S. residue of mod(Рn#), between which the P.I. sections are situated (with different length), represented as the alternances (array) of different amounts of different first primes ≤Рn – NOT residues mod(Рn#).

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

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 R3,2 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 = Рn, 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 -Рn = 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 + 2N:

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 2Nof the whole numbers, that is (Р3Р1) < 2Nwhole numbers.

2. The same primes are distributed as the loopback, pure two consequent primes at every P.I. sections, shorter than 2Nwhole numbers.

Proof. Every whole number = N is located within the squared two consequent primes: Р12 < N ≤ Р22 with: 2N>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 + 2N), 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 (Р3Р1) < 2Р1 whole numbers. As length of every section 2N> length of the section = 2Р1. Then from 1 to N + 2N– every:

(Р3Р1) < 2Nof 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 + 2N) 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 2Nwhole numbers >2Р1 whole numbers, that means, that at the P.I. section from 1 to N + 2Nat every P.I. section with length not exceeding 2Nwhole numbers, there is the loopback of primes, represented as the subgroup for two consequent primes.

Genuinely: Every P.I. section with length = 2Nof the whole numbers is located at the fractal -Р1# at the P.I. section. <Р22 as, Р12 < (N + 2N)Р22, with: 2N> 2Р1.

It is feasible, that there is a P.I. section with length = 2Nof the whole numbers, where there are no two primes, that is, two consequent primes are located at the P.I. section with length exceeding – 2Nof 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.

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Shcherbakov Aleksandr Gennadiyevich (September 29th 2020). Prime Numbers Distribution Line, Number Theory and Its Applications, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.92639. Available from:

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