Proof of the Conjecture (Together with the Proof of Polignac’s Conjecture for Cousin Primes) Marko Jankovic

To cite this version:

Marko Jankovic. Proof of the Twin Prime Conjecture (Together with the Proof of Polignac’s Conjec- ture for Cousin Primes). 2020. ￿hal-02549967v1￿

HAL Id: hal-02549967 https://hal.archives-ouvertes.fr/hal-02549967v1 Preprint submitted on 21 Apr 2020 (v1), last revised 18 Aug 2021 (v12)

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Marko V. Jankovic

Department of Emergency Medicine, Bern University Hospital “Inselspital”, and ARTORG Centre for Biomedical Engineering Research, University of Bern, Switzerland

Abstract. In this paper proof of the twin prime conjecture is going to be presented. In order to do that, the basic formula for prime numbers was analyzed with the intention of finding out when this formula would produce a twin prime and when not. It will be shown that the number of twin primes is infinite. The originally very difficult problem (in observational space) has been transformed into a simpler one (in generative space) that can be solved. The same approach is used to prove the

Polignac’s conjecture for cousin primes. The presented proof represents a small modification of the procedure that was used to prove the Sophie Germain primes conjecture.

1 Introduction

In number theory, Polignac’s conjecture states: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with the difference n [1]. For n = 2 it is known as twin prime conjecture.

Conditioned on the truth of the generalized Elliott-Halberstam Conjecture [2], in [3] it has been shown that there are infinitely many primes’ gaps that have a value of at least 6. In this paper gap 2 is analyzed. Gap 4 can be analyzed in a very similar manner. The problem is

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addressed in generative space, which means that prime numbers are not going to be analyzed directly, but rather their representatives that are used to produce them.

The proof is strategically very similar to the Sophie Germain primes conjecture proof that was recently presented in [4].

Outline of the proof: First, generalized Sophie Germain primes were defined. Then it was shown that twin primes are defined by the following formulas

푝푠 =6푘 − 1

푝푙 =6푘 + 1, 푘 휖푁.

Then numbers (ks) that cannot be used for the generation of twin primes are going to be analyzed. The number of those numbers is going to be compared with the number of numbers that are used for the generation of three groups of odd numbers: composite odd numbers, odd prime numbers whose safe relative is and odd prime numbers whose generalized safe relative is composite number (safe relative and generalized safe relative are going to be defined later in the text). The comparison will lead to the conclusion that the number of numbers that can be used for the generation of twin primes is infinite. It is important to notice that the number of numbers n, that are used for the generation of odd numbers bigger than 1 (2n+1, n ϵ N) and the number of the numbers k, that are used to produce numbers in the form 6k – 1 and 6k + 1 (k ϵ N), are the same.

Remark 1: Prime numbers 2 and 3 are in a sense special primes, since they do not share some of the common features of all other prime numbers. For instance, every , apart

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from 2 and 3, can be expressed in the form 6l + 1 or 6l - 1, where l ϵ N. So, in this paper most of the time, prime numbers bigger than 3 are analyzed.

Remark 2: In this paper any infinite number series in the form c1 * l ± c2 is going to be called a thread, defined by number c1. Here c1 and c2 are constants that belong to the set of natural numbers (c2 can also be 0, and usually is smaller than c1) and l represents an infinite series of consecutive natural numbers in the form (1, 2, 3, …). In the case when l is represented in the form 3*l1 - d, where d ϵ {0, 1, 2} and l1 represents an infinite series of consecutive natural numbers in the form (1, 2, 3, …), we call the thread c1 * (3*l1 - d)± c2 a sub-thread of the thread that is defined by c1. Obviously, every of those three sub-threads represent one third of the numbers that are represented by the thread defined by c1. In general case it is possible to define sub-threads for any natural number (instead of 3), but it is not going to be used in this paper.

It is also obvious that all threads defined by the same number represent the same number of numbers independently of the value of c2. It must be said that the thread that is defined as c1 * l - c2 has a potential to represent one number more than the thread defined by c 1 * l + c2, since it represents one number that is smaller than c1, while in the latter case it is not possible.

2 Sophie Germain primes

A prime p is a Sophie Germain prime if 2p + 1 is a prime too [5]. In that case the prime number 2p + 1 is called safe prime. Recently, it has been shown that infinite number of

Sophie Germain primes exists [4]. Here we will give a short recapitulation of the results that are of interest in this paper.

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Note: If p is prime number and 2p+1 is a composite number, the number 2p+1 is called a safe relative.

It is easy to check that any prime number (apart form 2 and 3) can be expressed in the form

6l+1 or 6s-1 (l, s ϵ N). Having that in mind, it is easy to conclude that numbers in the form 6l

+ 1, could never be Sophie Germain primes since their safe primes are in the form

2(6l + 1) + 1 = 12l + 3 = 3(4l + 1), and that is a composite number divisible by 3. Hence, the prime number that can potentially be a Sophie Germain prime must be in the form 6s – 1. The safe prime will then be in the form

6(2s) - 1.

We denote any composite number (that is represented as a product of prime numbers bigger than 3) with CPN5. A number in the form 6l + 1 is marked with mpl, while a number in the form 6s - 1 is marked with mps (l, s ϵ N). That means that any composite number CPN5 can be expressed in the form mpl x mpl, mps x mps or mpl x mps.

In [4], the composite numbers in mps form, as well as prime numbers in mps form whose safe relative is a composite number, were represented with 6k - 1 (k ϵ N). In that case the following equations must hold

푘 = (1)

푘 = , (2) ∙ where CPN5 is a composite number in the mps form.

From [4] we know that ks that cannot define a Sophie Germain prime are defined by the following equations

푘 = (6푥 −1)푦 + 푥 (3)

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(6푥 +1)푦 − , 푥 푖푠 푒푣푒푛 푘 = (4) (6푥 +1)푦 −3푥 − , 푥 푖푠 표푑푑

Equation (3) defines ks that produce composite numbers in mps form while ks defined by

(4) represent also the prime numbers whose safe relative is a composite number. Here has to be said that the following equation

(6푥 −1)푦 + , 푥 푖푠 푒푣푒푛 푘 = , (5) (6푥 −1)푦 −3푥 + , 푥 푖푠 표푑푑 is equivalent to (4) in the sense that it produces the same numbers as equation (4). That is shown in [4].

3 Generalized Sophie Germain primes

Here, the generalized Sophie Germain prime is going to be defined by the following statement:

A prime p is a generalized Sophie Germain prime if 2p ± (2i-1) is a prime too, where i ϵ N and i < p. The number 2p ± (2i-1) is called a generalized safe prime (GSP).

Note: If p is prime number and 2p ± (2i-1) is a composite number, the number 2p ± (2i-1) is called a generalized safe relative. That generalized safe relative is marked as GSR(±i).

A generalized Sophie Germain prime is marked as gSGP(i). Then, the original Sophie Germain prime is gSGP(1). In this paper we are interested in gSGP(-1), too. This means the following: a prime p is gSGP(-1) if 2p -1 is a prime too. Here, the number 2p-1 is marked as GSP(-1).

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Now, gSGP(-1) is going to be briefly analyzed. Using the same way of reasoning as in [4] it is easy to prove that gSGP(-1), as well as corresponding GSP(-1), must be in mpl form. If we represent the mpl form with 6l + 1 (l ϵ N) (following the same procedure like in [4]), it is easy to see that GSP(-1) must be in the form 6*(2*l) + 1. If we represent all composite numbers in the mpl form, as well as prime numbers in mpl form that have generalized safe relative, with 6k + 1 (k ϵ N), following the same procedure like in the [4], the following equations define ks that cannot be used for generation of gSGP(-1)

(6푥 −1)푦 − 푥 푘 = , (6) (6푥 +1)푦 + 푥

(6푥 +1)푦 + , 푥 푖푠 푒푣푒푛 푘 = , (7) (6푥 +1)푦 −3푥 + − 1, 푥 푖푠 표푑푑 (6푥 −1)푦 − , 푥 푖푠 푒푣푒푛 푘 = . (8) (6푥 −1)푦 −3푥 − + 1, 푥 푖푠 표푑푑

Equation (6) defines composite numbers in the mpl form, while equations (7) and (8) also define primes in the mpl form that have a generalized safe relative. Here we will make a conjecture that the number of gSGP(-1) is infinite. However, we are not going to analyze the problem here and, as it is going to be seen, it is not necessary to prove it in order to prove that number of twin primes is infinite (actually, once the twin prime conjecture is proved it is not difficult to prove that the number of gSGP(-1) is infinite).

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4 Twin primes

It is well known that every two consecutive odd numbers (psn, pln), between two consecutive odd numbers divisible by 3 (e.g. 9 11 13 15, or 39 41 43 45), can be expressed as

푝푠 =6푘 − 1

푝푙 =6푘 + 1, 푘 휖푁.

Twin prime numbers are obtained in the case when both psn, and pln are prime numbers. If any of the psn or pln (or both) are a composite number, then we cannot have twin primes. So, the strategy is to check in which cases (for which k) it would not be possible to have twin primes.

If psn represents a composite number, the following equation must hold

푘 = . (9)

Since CPN5 should be in the mps form, CPN5 can generally be expressed as a product mpl x mps, or

mpl = 6x + 1 and mps = 6y - 1(x, y ϵ N), which leads to

CPN5 = mpl x mps = 6(6xy - x + y) – 1, (10) or, due to symmetry

mpl = 6y + 1 and mps = 6x - 1, which leads to

CPN5 = mpl x mps = 6(6xy + x - y) – 1. (11)

If (10) or (11) is replaced in (9), forms of k that cannot produce the twin primes will be obtained. Those forms are expressed by the following equations

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푘 = (6푥 −1)푦 + 푥 (12a)

푘 = (6푥 +1)푦 − 푥, (12b) where x, y ϵ N. These equations are equivalent (they will produce the same numbers) and can be used interchangeably.

Using similar procedure, we can see that pln, that represents CPN5 number in the mpl form, will correspond to the composite number in the case

푘 = . (13)

In this case, CPN5 can be expressed in the form mpl1 x mpl2, or as mps1 x mps2. Two possibilities exist (x, y ϵ N):

mpl 1= 6x + 1 and mpl2 = 6y + 1, which leads to

CPN5 = mpl1 x mpl2 = 6(6xy +x + y) + 1, (14) or,

mps1 = 6x - 1 and mps2 = 6y - 1, which leads to

CPN5 = mps1 x mps2 = 6(6xy - x - y) + 1. (15)

When (14) and (15) are replaced in (13), together with (12b) the forms for all k that cannot produce a twin prime pair, are obtained. Those forms are expressed by the following equation

(6푥 +1)푦 − 푥 푘 = (6푥 −1)푦 − 푥, (16) (6푥 +1)푦 + 푥

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where x, y ϵ N. This is a sufficient and necessary condition for k, so that it cannot be used for the generation of twin primes. In other words, at least one of the twin odds generated by a k in any of the forms (16) will be a composite number, and if any of the odds generated by a k is a composite number, then k must be in one of the forms (16). Alternatively, it is possible to use the equation (12a) instead of (12b). In that case a different set of equations that produce the same numbers as the equation (16), is obtained. Here, a list of the k s(first 7) that cannot be presented in the form (16) and that generate all twin primes bigger than 3 and smaller than 100, is presented.

k 1 2 3 5 7 10 12

Twin prime 1 5 11 17 29 41 59 71

Twin prime 2 7 13 19 31 43 61 73

In order to prove that there are infinitely many twin prime pairs we need to prove that infinitely many natural numbers that cannot be expressed in the form (16), exist. First, the form of (16) for some values of x will be checked.

Case x=1: k = 5y - 1, k = 5y + 1, k = 7y + 1,

Case x=2: k = 11y - 2, k = 11y + 2, k = 13y + 2,

Case x=3: k = 17y - 3, k = 17y + 3, k = 19y + 3,

Case x=4: k = 23y - 4, k = 23y + 4, k = 25y + 4 = 5(5y +1) – 1,

Case x=5: k = 29y - 5, k = 29y + 5, k = 31y + 5,

Case x=6: k = 35y – 6 = 7(5y – 1) + 1, k = 35y + 6 = 5(7y + 1) +1, k = 37y + 6,

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Case x=7: k = 41y - 7, k = 41y + 7, k = 43y + 7,

Case x=8: k = 47y – 8, k = 47y + 8, k = 49y + 8 = 7(7y + 1) +1.

It can be seen that k is represented by the threads that are defined by prime numbers bigger than 3. From examples (cases x = 4, x = 6 and x = 8), it can be seen that if (6x - 1) or (6x +

1) represent a composite number, k that is represented by the thread defined by that number also has representation by the thread that is defined by one of the prime factors of that composite number. This can be proved easily in the general case, by direct calculation, using representations similar to (10) and (11). Here, only one case is going to be analyzed. All other cases can be analyzed analogously. In this case, assume that

(6푥 −1) = (6푙 +1)(6푠 −1) where (l, s ϵ N), and that leads to

푥 =6푙푠 − 푙 + 푠.

Considering that and using the following representation of k that includes the form (6x - 1)

푘 = (6푥 −1)푦 − 푥, the simple calculations leads to

푘 = (6푙 +1)(6푠 −1)푦 −6푙푠 + 푙 − 푠 = (6푙 +1)(6푠 −1)푦 − 푠(6푙 +1) + 푙, or

푘 = (6푙 +1)(6푠 −1)푦 − 푠 + 푙 which means

푘 = (6푙 +1)푓 + 푙

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and that represents the already existing form of the representation of k for the factor (6l+1), where

푓 = (6푠 −1)푦 − 푠.

It can be seen that all patterns for k can be represented by the thread defined by the prime number bigger than 3. Now, it is going to be proved that the number of natural numbers that cannot be represented by the models (16) is infinite.

. 5 Proof of the twin prime conjecture

Here, we are going to analyze the number of numbers that produce composite odd numbers, odd prime numbers that have GSR(1) and odd prime numbers have GSR(-1). It is well known that odd numbers a (bigger than 1) can be represented by the following formula

푎 =2푛 + 1, where n ϵ N. We know that the number of numbers n, that are used for the generation of odd numbers bigger than 1 and the number of the numbers k, that are used to produce numbers in the form 6k – 1 and 6k + 1 (k ϵ N), are the same.

From [4], it is known that if the form 2m+1, m ϵ N, represents odd numbers that are composite, the following equation holds

2푚 +1= (푖 +1)(푗 +1), where i1, j1 ϵ N. From [4] it is also known that the following equation holds

푖 푗 + 푖 + 푗 푚 = . 2

In order to have m ϵ N, it is easy to check that i1 and j1 have to be in the forms

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i1 = 2*i and j1 = 2*j,

where i, j ϵ N. From that, it follows that m must be in the form

푚 =2푖푗 + 푖 + 푗. (17)

Once more, all composite odd numbers are presented by 2m+1, where m is given by (17).

Now, we have to check when an odd number is in the mps or mpl form. An odd number represented by 2m+1, m ϵ N, is in the mps form when m is in the form m= 3w-1, w ϵ N, and it is in the mpl form when m is in the form m= 3w, w ϵ N.

Equations (4), (7) and (8) define the values for k that produce numbers that have GSR(1) or

GSR(-1). That means that following equations define values m that will produce odd numbers that are composite and primes that have GSP(1) or GSP (-1)

3푥 (6푥 +1) ∙ (3푦) − − 1, 푥 푖푠 푒푣푒푛 푚 = 2 푥 +1 (6푥 +1) ∙ (3푦) −9푥 −3 − 1, 푥 푖푠 표푑푑 2

(6푥 −1) ∙ (3푦)−3 , 푥 푖푠 푒푣푒푛 푚 = . (18) (6푥 −1) ∙ (3푦)−9푥 −3 + 3, 푥 푖푠 표푑푑 푥 (6푥 +1) ∙ (3푦) +3 , 푥 푖푠 푒푣푒푛 2 푚 = 푥 +1 (6푥 +1) ∙ (3푦) −9푥 +3 − 3, 푥 푖푠 표푑푑 2

If (18) is checked for some values of x the following equations are obtained

Case x=1: m = 7(3y - 2) + 1, m = 5(3y - 1) - 4, m = 7(3y - 2) + 5

Case x=2: m = 13(3y) + 2, m = 11(3y) - 3, m = 13(3y) + 3

Case x=3: m = 19(3y - 2) + 3, m = 17(3y - 2) + 4, m = 19(3y -2) + 14

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Case x=4: m = 5(3(5y)-1) - 2, m = 23(3y) - 6, m = 5(3*(5y+1) - 1) - 4 (in the first thread the equivalence of (4) and (5) is used)

Case x=5: m = 31(3y-2) + 7, m = 29(3y-1)-22, m = 31(3y - 2) + 23

Case x=6: m = 37 (3y-1) +27, m = 5(3(7y) -1) - 4, m = 37(3y) + 9

Case x=7: m = 43(3y-2)+10, m = 41(3y-1)-31, m = 43(3y-2) + 32

Case x=8: m = 7(3(7y)-2) + 1, m = 47(3y)-12, m = 7(3*(7y+1) -1) – 2.

It can be seen that all sub-threads are defined by prime numbers bigger than 3. If the sub- thread is defined by some composite number it can be shown that such a sub-thread can be represented by the sub-thread defined by some prime factor of that composite number (like in the cases of the numbers k that define numbers that cannot produce twin primes).

From [4], it is known that m defined by (17) is represented by the threads defined by odd prime numbers. Beside threads that are defined by prime numbers in the mps and mpl form,

(17) also defines a thread that is defined by prime number 3. That thread will actually produce one sub-thread for all threads defined by primes bigger than 3. This is easy to check.

For instance the thread 3y+1, y ∈ N, will produce numbers 3*(5z)+1 = 5*(3z) +1,

3*(7z)+1= 7*(3z) +1, (z ∈ N), and so on, The thread defined by number 3 represents all numbers that produce remainder 1 if they are divided by 3. It is important to notice that no sub-threads defined by (18) will represent a number that has remainder 1 when it is divided by 3.

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Now we are going to compare the number of sub-threads that are defined by equation (16) on one side and on the other side the number of sub-threads defined by equations (17) and

(18).

Equation (16) defines 3 sub-threads defined by every prime in the mps form, and 6 sub- threads defined by every prime in the mpl form,

Equation (17) defines 3 sub-threads defined by every prime number in the mps form, 3 sub- threads for every prime number in the mpl form and thread defined by number 3. As it has already been said, it is easy to see that a thread defined by three makes one sub-thread for every prime in the mpl form (and many others that we are going to ignore). Equation (18) defines 2 sub-threads defined by each prime in the mpl form and 1 sub-thread defined by each prime in the mps form. So, equations (17) and (18) define 4 sub-threads in the mps form and 6 sub-threads in the mpl form (and many others that are currently not of interest).

It can be clearly concluded that there are more numbers defined by (17) and (18), than numbers defined by (16). Now, it will be assumed that number of gSGP(-1) is finite, which is the worst case for our analysis.. That means that numbers that cannot be represented by

(17) and (18), are numbers that create odd numbers that are Sophie Germain primes and some finite number of gSGP(-1). Since it is known that the number of the Sophie Germain primes is infinite [4], it can safely be concluded that the number of twin primes is infinite, too. That completes the proof of the twin prime conjecture.

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6 Proof that the number of cousin primes is infinite

The cousin prime numbers are prime numbers with the gap 4. It is clear that cousin primes represent pairs of odd numbers that surround odd number divisible by 3 (e.g. (7 9 11), or

(13 15 17)). A pair can only represent a cousin primes if both of those numbers are primes.

If the pair of odd numbers that surround the odd number divisible by 3 are denoted as pln =

6k+1 and psm = 6(k+1)−1 = 6k+5, k ∈ N, these numbers can represent cousin primes only in the case when both pln and psm are prime numbers. If any of the pln or psm (or both) is a composite number, then we cannot have cousin primes.

The strategy is to check in which cases (for which k) it would not be possible to have cousin primes.

If psm represents a composite number the following equation must hold

6k + 5 = CPN5, where CPN5 represents a composite number in the mps form. After some elementary calculations, the following equation can be obtained

푘 = = −1. (19)

Since CPN5 should be in the mps form, CPN5 can generally be expressed as a product mpl × mps, or

mpl = 6x + 1 and mps = 6y − 1(x, y ∈ N), which leads to

CPN5 = mpl × mps = 6(6xy − x + y) − 1, (20) or, due to symmetry

mpl = 6y + 1 and mps = 6x − 1,

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which leads to

CPN5 = mpl × mps = 6(6xy + x − y) − 1. (21)

Following a similar procedure, it can be seen that pln, that represents a CPN5 number in the mpl form, will correspond to the composite number in the case

푘 = . (22)

In this case, CPN5 can be expressed in the form mpl1 × mpl2 or as mps1 × mps2, and we will have two possibilities (x, y ∈ N)

mpl1 = 6x + 1 and mpl2 = 6y + 1, which leads to

CPN5 = mpl1 × mpl2 = 6(6xy + x + y) + 1, (23) or

mps1 = 6x − 1 and mps2 = 6y − 1, which leads to

CPN5 = mps1 × mps2 = 6(6xy − x − y) + 1. (24)

So, if we replace (20) (equivalently (21)) in (19), and (23) and (24) in (22) the forms for all k that will not produce a cousin prime pair are obtained. Those forms are expressed by the following equation

(6푥 +1)푦 − 푥 −1 푘 = (6푥 −1)푦 − 푥 , (25) (6푥 +1)푦 + 푥 where x, y ∈ N. This is sufficient and necessary condition for k, so that it cannot be used for the generation of cousin primes.

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Now, using the same method like in the case of the twin prime conjecture, it can be proved that infinitely many natural numbers that cannot be presented in the form (16) exist, and that completes the proof that the number of cousin primes is infinite.

References

[1] de Polignac, A. (1849). Recherches nouvelles sur les nombres premiers. Comptes

Rendus des S´eances de l’Acad´emie des Sciences.

[2] Neale, V. (2017). Closing the Gap. Oxford, U. K., Oxford University Press, p. 141-144.

[3] D.H.J. Polymath. Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences. 1(12). arXiv:1407.4897 (2014).

[4] M. Jankovic. A proof of Sophie Germain primes conjecture. 2020. ⟨hal-02169242v3⟩

[5] T. Agoh. On Sophie Germain primes, Tatra Mt. Math. Publ 20(65) (2000), 65-73.

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