Proof of the Twin Prime 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) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 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 1 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 composite number 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 prime number, apart 2 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. 3 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) 4 (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). 5 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 = .