Abstract

It is well known that every prime number has the form or We will call the generator of Twin primes are distinghuished due to a common generator for each pair. Therefore it makes sense to search for the Twin Primes on the level of their generators. This paper present a new approach to prove the Twin Prime Conjecture by a sieve method to extract all Twin Primes on the level of the Twin Prime Generators. We define the --numbers as numbers for which holds that and are coprime to the prime By dint of the average distance between the --numbers we can prove the Twin Prime Conjecture indirectly.

Notations

We will use the following notations:

and

1. Introduction

The question on the infinity of the twin primes keeps busy many mathematicians for a long time. 1919 V. Brun ³ had proved that the series of the inverted twin primes converges while he had tried to prove the Twin Prime Conjecture. Several authors worked on bounds for the length of prime gaps (see f.i. ^{4, 5, 6}). 2014 Y. Zhang ⁷ obtained a great attention with his proof that there are infinitely many consecutive primes with a distance of 70,000,000 at most. With the project ''PolyMath8'' this bound could be lessened down to 246 respectively to 12 assuming the validity of the Elliott-Halberstam Conjecture ⁸.

We present in this paper another approach as in the most works on this topic. We tranfer the looking for twin primes to the level of their generators because each twin prime has a common generator.

2. Twin Prime Generators

It is well known that every prime number has the form or We will call the generator of Twin primes are distinghuished due to a common generator for each pair. Therefore it makes sense to search for the twin primes on the level of their generators.

Let be

(2.1)

an over defined function, the generator of the pair .

A number is a member of if as well as are primes. This is true if the following statement holds.

Theorem 1. A number is a member of if and only if there is no with where one of the following congruences holds:

(2.2)

(2.3)

Proof.

A. therefore is

If (2.2) is true then there is an with

For (2.3) the proof will be done with :

B. , therefore is : We go the same way with (2.2) and as well as (2.3) and :

With these it's shown that if the congruences (2.2) or (2.3) are valid. They cannot be true both because they exclude each other.

If on the other hand , then is or no prime. Let be and . Then we have

For we have

The other both cases we can handle in the same way. Therefore either (2.2) or (2.3) is valid if

If we consider that the least proper divisor of a number or is less or equal to than in the congruences (2.2) and (2.3) can be further limited by

Henceforth we will use the letter for a general prime number and if we describe an element of a sequence of primes. With as the -the prime number^{1} and as the number of primes we have with

(2.4)

for a proofable system of criteria to exclude a number as not being a member of .

3. The Twin Sieve

The congruences in (2.4) can be combined in the following way:

(3.1)

because if then there is a number with . Squared this produces and we get . This results in a system of sieves with sieve functions for which hold for

(3.2)

Obviously is a periodical function in with a period length of . We'll call the sieve represented by as For the system of the sieves we'll build the aggregate sieve functions

(3.3)

Because the value set of consists of positive integers between and , and have rational values between and .

A number will be “sieved” by if and only if . With (3.3) in this case also is In contrast to the sieve of ERATOSTHENES in our sieve the exclusion of a number will be not controlled by but by

Let be

(3.4)

For “works'' the sieve i.e. is the origin of the sieve Every sieve has up from in every --period just positions with and two positions with , once if (2.2) and on the other hand if (2.3) is valid. We speak about - and -bars of the sieve From (2.2) and (2.3) it is easy to see that the distance between an - and a -bar is .

It is and therefore . Then

(3.5)

is the least number which meets this relation. It is easy to prove that for every prime holds that is an integer divisible by .

Theorem 2. Every sieve with starts at its origin with a sieve bar and we have .

Proof. We substitute by . With this and (3.5) holds

We see that starts for with an -bar (2.2) and in the other case with a -bar(2.3).

For every the local position in the sieve relative to the phase start^{2} can be determined by the position function :

(3.6)

Between the sieve function and the position function there is the following relationship:

(3.7)

Obviously is if and only if (-bar) or (-bar).

4. The Permeability of the Sieves S₃×…×S_n

For every x in the interval

(4.1)

persists constant on the value . The length of this interval^{3} will be denoted as . It is depending on the distance between successive primes. Since they can only be even, we have with a = 2,4,6,…

(4.2)

On the other hand it results because of ( ², p. 188)

(4.3)

The congruences from (3.6)

(4.4)

meet the requirements of the Chinese Remainder Theorem (see ¹, p. 89). Therefore it is modulo uniquely resolvable. With

(4.5)

it's ^{4} uniquely resolvable. Therefore the sieves have the period length and for the aggregate sieve function holds:

Definition 4.1. A positive integer will be called an “--number” if both and are coprime^{5} to . In this case is .

Let be

the interval of the period of the sieves We'll denote it henceforth as period section. Evidently is for all .

The values of the function are the numbers . Two of them result in the exlcuding of and don't. Therefore by working of the sieves we have

(4.6)

--numbers in . If a lot of them are in , they are members of because the sieves here are working only. The relation between (4.6) and the period length of (4.4) results in

(4.7)

as a measure of the mean “permeability” of working of the sieves or as the density of the --numbers in Obviously is a strong monotonously decreasing function. Its inversion

(4.8)

discribes the average distance between the --numbers in their period section.

Theorem 3. The density of the --numbers in their period section is lower bounded by

Proof. Let be and

Because all primes are odd numbers it holds for ^{6}. All factors of are less than 1. It results

By inversion of this relationship, we obtain for the average distance

(4.9)

Under consideration of (4.2) we obtain furthermore^{7}

(4.10)

This means that the avarage distance between -numbers remains ever less than the half of the length of , the interval where -numbers are twin prime generators.

5. The Sieve Process on Average

The intervals defined by (4.1) cover the positive integers gapless and densely. It is

They are the beginnings of the period sections of the --numbers. Hereafter let's say A-sections to the intervals . Every --number which lies in an A-section is a twin prime generator (see above). In contrast to the A-sections the period sections overlap each other very densely. So the period section reachs over 1739 A-sections up to the beginning of the period section and the next over 7863 A-sections up to the beginning of

Theorem 4. Each origin cannot be located at the beginning of any period of the sieves for Therefore it holds for

Proof. The equation

is for no primes solvable, because of

Vice versa holds that every period section starts always inside of the previous period section nearby to its origin because (see (4.3) also)

Let be

With these we can show the recursive structure of the period sections

(5.1)

We can clearly see that the period section overlaps up to the end of and has much space for A-sections with .

The one consequence of this dense overlapping of the period sections is that a plurality of the --gaps from the period section persist constant as also --gaps for but in a shifted position relative to their origin (see Theorem 4 and (5.1)).

On the other hand this dense overlapping guarantees that extreme anomalies of the distribution of the --numbers cannot occur.

For the quantity of the --numbers in is corresponding with (4.6)

In the --numbers are spread^{8} over positions. According to (5.1) we have --numbers in In comparison between them and the --numbers resulting from the working of the sieve we see

(5.2)

We loose by the working of in the period section just potential generators of twin primes. In other words, the sieve has `beating bars'' in At these positions holds

(5.3)

Only the beating bars let grow the gaps by exclusion of the --number between two --gaps to one --gap. By the working of the sieve we obtain the following sieve balance “on average'':

The distances between the --numbers persist unchanged at on average except of those --numbers which are met by the beating bars of the sieve . Thereby a distance occurs between the adjacent --numbers on average:

Therefore even the gaps between the --numbers (--gaps) which result from the beating bars persist less than on average because

(5.4)

6. Proof of the Twin Prime Conjecture

The proof will be done indirectly. We assume that there is only a finite number of twin primes and therefore there is only a finite number of twin prime generators. Let be the greatest one. It lies in the A-section with , the beginning of the period section In the subsequent A-sections with consequently there cannot be any twin prime generators and therefore no --numbers. But then we have --gaps with lengths in all (infinitely many) period sections for .

Because

Ÿ all period sections are very densely overlapped and therefore extreme anomalies of the distribution of the --numbers cannot occur,

Ÿ the average distances between the --numbers are less than ,

Ÿ and even the --gaps which are generated by beating bars of the sieves are less than on average,

therefore it is not possible to have for all only period sections with --gaps at their beginnings which are all greater than .

Therefore the proof assumtion cannot be valid and thus the Twin Prime Conjecture must be true.

Notes

1. It is p₁ = 2.

2. For the phase start is and else it is

3. Really is Henceforth all intervals will be defined as sections of the number line.

4. It is with the primorial

5. Then is

6. For is

7. We can even prove that and for .

8. It is easy to prove that the --numbers in their period section are symmetrically distributed around and Nevertheless the distribution is non-uniform.

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