The Twin Prime Conjecture Ricardo Barca To cite this version: Ricardo Barca. The Twin Prime Conjecture. 2018. hal-01882333v2 HAL Id: hal-01882333 https://hal.archives-ouvertes.fr/hal-01882333v2 Preprint submitted on 13 Nov 2020 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. Public Domain THE TWIN PRIME CONJECTURE Ricardo G. Barca Abstract Consecutive primes which differ only by two are known as twin primes; the function π2(x) describes the distri- bution of the twin primes in the set of natural numbers. The twin prime conjecture states that there are infinitely many primes p such that p + 2 is also prime. At the present time, the knowledge about twin primes comes mostly from sieve methods. Using a formulation of the sieve of Eratosthenes supported on a sequence of k-tuples of re- mainders we obtain a lower bound for the number of twin primes in [1; x]. Thus, we prove that π2(x) ! 1 as x ! 1. 1 Introduction The twin prime conjecture is one of the problems in number theory that still remains without resolution. Twin prime conjecture: There exist infinitely many primes p such that p + 2 is a prime. In 1849, the french mathematician de Polignac made the statement (more general than twin prime conjecture) that for every positive even integer k there exist infinitely many primes p such that p + k is also a prime [6]. For k = 2 we have the twin prime conjecture. The most significant result for twin primes is due to Viggo Brun [3], who proves, in 1915, that the sum of the reciprocals of the odd twin primes, B = (1=3 + 1=5) + (1=5 + 1=7) + (1=(11) + 1=(13)) + (1=(17) + 1=(19)) + ::: converges to a number known as the Brun's constant. This outcome was the first use of the Brun sieve, and it was the beginning of modern sieve theory. Since the series of all prime reciprocals diverges to infinity, the result from Viggo Brun say us that twin primes are very scarce. On April 17, 2013, the american mathematician (born in Shangai) Yitang Zhang claims that there are infinitely many pairs of prime numbers that differ by 70 million or less [4]. For any m ≥ 1, let Hm denote the quantity 7 Hm = lim infn!1 (pn+m − pn), where pn denotes the n-th prime. In his paper, Zhang proved that H1 ≤ 7 × 10 ; the twin prime conjecture is equivalent to the assertion that H1 is equal to two. After the appearance of Zhang's paper, improvements to the upper bound on H1 were made. In addition, a Polymath project was made up to examine the arguments of Zhang [5]; one year after Zhang's announcement, the bound has been reduced to 246. Let x > 0 be a positive real number; let π2(x) be the number of twin primes p and p + 2 such that p ≤ x. In this paper we prove (Main Theorem) the following: π2(x) ! 1 as x ! 1. One of the principal ways of addressing problems like the twin prime conjecture or the Goldbach's conjecture have been through the use of sieve methods. The Sieve theory has developed along of the last century, starting from the Viggo Brun methods. For the sake of completeness and convenience of the reader, in this paper we shall repeat some explanations and arguments already given in [9]. Also, we use concepts and tools developed in [9], as well as the notations defined in this paper, unless we specifically state otherwise. We begin by defining the sieve of Eratosthenes, using the notation of the book by Cojocaru andp Ram Murty [2]. Let A = fn 2 Z+ : n ≤ xg, where x 2 R; x > 1, and let P be the sequence of all primes. Let z = x, and Y P (z) = p: p2P p<z Now, to each p 2 P; p < z, we associate the subset Ap of A , defined as follows: Ap = fn 2 A : n ≡ 0 (mod p)g. Then, when we sift out from A all the elements of every set Ap, the unsifted membersp of A are the integers that are not divisible by primes of P less than z; that is to say, any integer remaining in [ x; x) is a prime. Furthermore, if d is a squarefree integer such that djP (z), we define the set \ Ad = Ap: pjd Thus, from the inclusion{exclusion principle, we obtain 1 X S(A ; P; z) = µ (d) jAdj ; (1) djP (z) where µ(d) is the M¨obiusfunction. Moreover, from (1), the Legendre formula can be derived X X jxk S(A ; P; z) = µ (d) jAdj = µ (d) : d djP (z) djP (z) p The sieve of Eratosthenes is useful for findingp the prime numbers between x and x; for our purposes, we need first to isolate the subset of twin primes between x and x and then to obtain a lower bound for the size of this subset, for every x sufficiently large. However, we can not hope to find a suitable lower bound for the size of this set using the usual sieve methods, due to a well-known phenomenon in sieve theory, called the `parity barrier' or the `parity problem' (see [9, Introduction, Subsection 1.1]). So far, all attempts to solve the twin prime problem by the usual sieve techniques did not have the expected success. For these reasons, the strategy used in this paper differs quite a lot from the usual approach in sieve theory. n 2 3 5 7 n 2 3 5 7 1 1 1 1 1 31 1 1 1 3 2 0 2 2 2 32 0 2 2 4 3 1 0 3 3 33 1 0 3 5 4 0 1 4 4 34 0 1 4 6 5 1 2 0 5 35 1 2 0 0 6 0 0 1 6 36 0 0 1 1 7 1 1 2 0 37 1 1 2 2 8 0 2 3 1 38 0 2 3 3 9 1 0 4 2 39 1 0 4 4 10 0 1 0 3 40 0 1 0 5 11 1 2 1 4 41 1 2 1 6 12 0 0 2 5 42 0 0 2 0 13 1 1 3 6 43 1 1 3 1 14 0 2 4 0 44 0 2 4 2 15 1 0 0 1 45 1 0 0 3 16 0 1 1 2 46 0 1 1 4 17 1 2 2 3 47 1 2 2 5 18 0 0 3 4 48 0 0 3 6 19 1 1 4 5 49 1 1 4 0 20 0 2 0 6 50 0 2 0 1 21 1 0 1 0 51 1 0 1 2 22 0 1 2 1 52 0 1 2 3 23 1 2 3 2 24 0 0 4 3 25 1 1 0 4 26 0 2 1 5 27 1 0 2 6 28 0 1 3 0 29 1 2 4 1 30 0 0 0 2 Figure 1 Let P be the sequence of all primes; and given pk 2 P, let mk = p1p2p3 ··· pk. From now on, and throughout 2 this paper, for convenience, we take x to be a real number greater than p4 = 49, unless we specifically state otherwise. p 2 2 Note that if pk is the greatest prime less than x, every real number x > 49 satisfies pk < x ≤ pk+1 < mk. 2 Now, the formulation of the sieve of Eratosthenes using the classic sieve theory notation does not allow us to discriminate between the twin primes and the other primes, for counting only the first ones. In this paper we propose to use another formulation for this kind of sieves, using a sequence of k-tuples, which is able to show all the details of the sifting process (see [9, Introduction, Subsectionp 1.3]). Let pk be the greatest prime less than x. In the formulation of the sieve of Eratosthenes by means of a sequence of k-tuples, the index k is the corresponding to pk, and A is the set of integers that lie in the interval [1; x]. The rules for selecting remainders are the following: In the k-tuples of the sequence, all the zeroes are defined as selected remainders. Therefore, within the periods of every sequence of remainders modulo ph (a given column of the matrix), the element 0 is always a selected remainder. We more formally define the formulation of this sieve using a sequence of k-tuples in Section 2. p Remark 1.1. Note that given a k-tuple whose index ispn ≤ x, if n ≡ 0 (mod p) for at least one p < x, then it is a prohibited k-tuple, and if n 6≡ 0 (mod p) for every p < x, then it is a permitted k-tuple. p Clearly, the indices of the permitted k-tuples lying inp the interval [ x; x) are prime numbers. Therefore, the indices of a pair of permitted k-tuples that lie in the interval [ x; x) and differ by two form a pair of twin primes.