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Proof of Polignac's Conjecture for Gaps Bigger Than Four

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Proof of Polignac's Conjecture for Gaps Bigger Than Four Proof of Polignac’s Conjecture for gaps bigger than four Marko Jankovic To cite this version: Marko Jankovic. Proof of Polignac’s Conjecture for gaps bigger than four. 2021. hal-03170059v3 HAL Id: hal-03170059 https://hal.archives-ouvertes.fr/hal-03170059v3 Preprint submitted on 9 May 2021 (v3), last revised 5 Sep 2021 (v4) 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. Proof of Polignac's Conjecture for gaps bigger than four Marko V. Jankovic Institute of Electrical Engineering “Nikola Tesla”, Belgrade, Serbia, Department of Emergency Medicine, Bern University Hospital “Inselspital” and ARTORG Center for Biomedical Engineering Research, University of Bern, Switzerland Abstract In this paper proof of the Polignac's Conjecture for gaps bigger than four is going to be presented. The proof represents an extension of the proof of the twin prime conjecture. It will be shown that primes with a gap of size bigger than four could be obtained through two stage sieve process, and that will be used to obtain a reasonable estimation of their numbers. 1 Introduction In number theory, Polignac's conjecture states: For any positive even number g, there are infinitely many prime gaps of size g. In other words: there are infinitely many cases of two consecutive prime numbers with the difference g [1]. The conjecture was proved for gaps bigger than 244 in [2]. Recently, the conjecture was proved for gaps of the size 2 and 4 [3]. The problem was addressed in generative space, which means that prime numbers were not analyzed directly, but rather their representatives that can be used to produce them. This paper represents an extension of the previous work [3]. Basically, three groups of gaps bigger than 4 exist: the gaps of the size 6k, the gaps of the size 6k+2 and gaps of the size 6k+4, k ϵ N. In the text that follows we mark the prime numbers in the form 6k – 1 as mps primes and prime numbers in the form 6k +1 as mpl primes, k ϵ N. The gaps of the size 6k could be related to the prime pairs in both (mps, mps) and (mpl, mpl) form. The gaps in the form 6k + 2 can only be related to the pair of primes in (mps, mpl) form, while gaps in the form 6k + 4 can only be related to the pair of primes in (mpl, mps) form. In other words there is not a single prime in mpl form that has consecutive prime that is 6k + 2 apart, and there is not a single prime in mps form that has consecutive prime 6k + 4 apart. It is trivial to show that by simple calculation. Although last two groups are different, they can be treated analogously. So, here two different cases are going to be analyzed in order to explain how conjecture can be proved in general case. First it will be shown that exist an infinite number of primes with gaps of the size 6 (also known as sexy primes). After that, it will be shown that exists an infinite number of primes with the gap of size 8 (we mark them as 8-primes). Than the proof can easily be generalized for the gaps of any size g. It will be shown that primes with gap of a size g (g-primes) could be generated by two stage process, and that will be used to generate formula for the number of g-primes. It will be shown that exist lower bound for the number of g-primes smaller than some natural number n, n ϵ N, and that will be used to show that overall number of g-primes is infinite. Remark 1: In this paper any infinite series in the form c1·l ± c2 is going to be called a thread defined by number c1 (in literature these forms are known as linear factors – however, it seems that the term thread is probably better choice in this context). Here c1 and c2 are numbers that belong to the set of natural numbers (c2 can also be zero and usually is smaller than c1) and l represents an infinite series of consecutive natural numbers in the form (1, 2, 3, …). 2 Proof of the sexy prime conjecture It is well known that all prime numbers can be expressed in one of the following form psk = 6k - 1 plk = 6k + 1, k ϵ N. As it was already explained, we will call numbers psk - numbers in mps form and numbers plk - numbers in mpl form. If two consecutive prime numbers have the gap of the size 6 it is clear that both of those numbers have to be in mps form or both have to be in mpl form. In this paper we will only analyze the case of sexy primes in mps form. The case of sexy primes in mpl form can be analyzed analogously. Here we are going to present a two stage process that can be used for generation of the sexy primes in mps form. In the first stage we are going to produce prime numbers by removing all composite numbers from the set of natural numbers. In the second stage, we are going to remove all twin primes, all prime numbers in mpl form that are left, and all prime numbers in mps form that have an bigger sexy odd neighbor (odd number that has gap 6 with the prime of interest) that is a composite number. At the end, only the prime numbers in the mps form, that represent the smaller number of a sexy prime pair, are going to stay. Their number is approximately the half of the number of sexy primes in mps form, or one quarter of all sexy primes (although a little bit bigger due to existence of sexy triplets and quadruplets – in that case only the last number in the series is going to be removed from the set; here, it is considered that number of triplets and quadruplets sexy primes is very small comparing to the number of sexy primes). It is going to be shown that that number is infinite. Second half of the sexy primes is generated with similar procedure that takes care about sexy primes in mpl form. STAGE 1 Prime numbers can be obtained in the following way: First, we remove all even numbers (except 2) from the set of natural numbers. Then, it is necessary to remove the composite odd numbers from the rest of the numbers. In order to do that, the formula for the composite odd numbers is going to be analyzed. It is well known that odd numbers bigger than 1, here denoted by a, can be represented by the following formula a = 2n +1, where n ϵ N. It is not difficult to prove that all composite odd numbers a can be represented by the c following formula ac=2(2i j+i+ j)+1=2((2 j+1)i+ j)+1. (1) where i, j ϵ N. It is simple to conclude that all composite numbers could be represented by product (i + 1)(j + 1), where i, j ϵ N. If it is checked how that formula looks like for the odd numbers, after simple calculation, equation (1) is obtained. This calculation is presented here. The form 2m + 1, m ϵ N will represent odd numbers that are composite. Then the following equation holds 2 m+1=(i 1+1)( j1+1) , where i , j ϵ N. Now, it is easy to see that the following equation holds 1 1 i j +i + j m= 1 1 1 1 . 2 In order to have m ϵ N, it is easy to check that i and j have to be in the forms 1 1 i1 = 2i and j1 = 2j, where i, j ϵ N. From that, it follows that m must be in the form m = 2ij + i + j = (2i + 1) j + i. (2) When all numbers represented by m are removed from the set of odd natural numbers bigger than 1, only the numbers that represent odd prime numbers are going to stay. In other words, only odd numbers that cannot be represented by (1) will stay. This process is equivalent to the sieve of Sundaram [4]. Let us denote the numbers used for the generation of odd prime numbers with m2 (here we ignore number 2). Those are the numbers that are left after the implementation of Sundaram sieve. The number of those numbers that are smaller than some natural number n, is equivalent to the number of prime numbers smaller than n. We denote with π(n) number of primes smaller than n, than the following equation holds n π(n)≈ . ln (n) From [6] we know that following holds n π(n)> , n⩾ 17. (3) ln(n) STAGE 2 What was left after the first stage are prime numbers. With the exception of number 2, all other prime numbers are odd numbers. All odd primes can be expressed in the form 2n + 1, n ϵ N. It is simple to understand that their bigger sexy odd neighbor must be in the form 2n + 7, n ϵ N.
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