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Towards Proving the Twin Prime Conjecture Using A Towards Proving the Twin Prime Conjecture using a Novel Method For Finding the Next Prime Number PN+1 after a Given Prime Number PN and a Refinement on the Maximal Bounded Prime Gap Gi Reema Joshi Department of Computer Science and Engineering Tezpur University Tezpur, Assam 784028, India [email protected] Abstract This paper introduces a new method to find the next prime number after a given prime P . The proposed method is used to derive a system of inequali- ties, that serve as constraints which should be satisfied by all primes whose successor is a twin prime. Twin primes are primes having a prime gap of 2. The pairs (5, 7), (11, 13), (41, 43), etcetera are all twin primes. This paper envisions that if the proposed system of inequalities can be proven to have infinite solutions, the Twin Prime Conjecture will evidently be proven true. The paper also derives a novel upper bound on the prime gap, Gi between Pi and Pi+1, as a function of Pi. Keywords: Prime, Twin Prime Conjecture, Slack, Sieve, Prime Gap 2010 MSC: 11A41, 11N05 arXiv:2004.14819v2 [math.GM] 6 May 2020 1. Introduction A positive integer P > 1 is called a prime number if its only positive divisors are 1 and P itself. For example, 2,3,7,11,19 and so on. There are in- finitely many primes. A simple yet elegant proof of this proposition was given by Euclid around 300 B.C. One interesting question would be, "does there exist a method for generating successive prime numbers along the infinite pool of primes?". The answer is yes, and is elaborated in a very well known method for the same, known as the Sieve of Eratosthenes, an algorithm for generating prime numbers up to a given upper bound N. The complexity of the Sieve of Eratosthenes is O(N log log N). There are several variants of the Sieve method, which reduce the asymptotic complexity down to O(N), like the Sieve of Sundaram[1], Sieve of Atkin[2], and different Wheel Sieve methods. This paper proposes a simple method which can accurately calculate the next prime number after a given prime. It may be mentioned beforehand that this method is not very time-efficient(O(N 2)) as compared to existing methods, though its space complexity is O(N). This does not make it favourable for prime number generation. However, the goal of this paper is to use the method to revise the upper bound on the prime gap function, Gi, and ana- lyze significant properties of twin primes. As mentioned earlier, the condition for (Pi,Pi+1) to be twin primes is Pi+1 = Pi +2. The Twin Prime Conjecture asserts the infinitude of twin primes and has two different versions. This paper elaborates on the first, as mentioned in [6], and given below: Twin Prime Conjecture: There exist infinitely many primes P such that P +2 is a prime. In other words, lim (Pi+1 Pi)=2. n→∞ − Though mathematicians hypothesize that the conjecture may well be true[13], it has not been successfully proven yet. The distribution of primes seems arbitrary, in some cases, intervals of 2 are seen, for example, the primes 5 and 7 have a gap of 2, while in other cases there are arbitrarily long gaps between consecutive primes[9]. The Prime Number Theorem places a bound on the prime counting function. The prime counting function, denoted by πN counts the number of primes less than or equal to N. Gauss in 1792, stated that, N πN . ≈ log N This means for a large enough N, the probability that any number not ex- 1 ceeding N is prime is close to ( log N ). The theorem was independently proven by Jacques Hadamard and Charles Jean de la VallÃľe Poussin in 1896. 2 A detailed mention regarding the bounds on twin primes can be found in [7]. After the conjecture gained attention, several Sieve based methods have been used at attempts to prove it[11]. Zhang, in 2014, made a significant breakthrough along the lines of the Twin Prime Conjecture[15]. His paper stated that, 7 lim (Pn+1 Pn) < 7 10 . (1) n→∞ − × This implies that there are an infinite number of primes, Pn such that the gap between two consecutive primes is less than 70 million. In this paper, how- ever, a different approach has been used towards a proof of the Twin Prime Conjecture, based on the proposed method for calculating the successor of a given prime, P . The goals of this paper are summarized below: 1. To present a novel method for finding Pi+1 given Pi. 2. To provide a definite upper bound on the prime gap, Gi as a function th of the i prime, Pi. 3. To use the proposed method for setting constraints that should be mandatorily fulfilled by all prime numbers Pi whose successor prime is its twin, that is, Pi+1 = Pi +2. 2. Bounds on the Prime Gap, Gi A prime gap is the difference between two successive primes. If the length of the gap is n, it implies that there is a sequence of n 1 successive com- − posite numbers between Pi and Pi+1. So, Gi = Pi+1 Pi = n, − where Gi represents the prime gap. n is always even, except in the case of (2, 3) where n =1. We can easily infer that G1 =1, G2 =2 ,G3 =2, G4 = 11 and so on. Also, the sequence, S, S = n!+2, n!+3, n!+4, ..., n!+ n is a run of n 1 consecutive composite integers, n!+ x (n!+ x being divisible − by x). The above equation is evidently part of some prime gap, Gi, such that Gi n. As stated by the Prime Number Theorem, the average prime gap ≥ increases with the natural logarithm of the prime number, P . The ratio 3 Gi is called the merit of the gap Gi. The largest known prime gap is log Pi 1113106 digits long, and has merit of 25.90. It was discovered by J. K. Andersen, M. Jansen and P. Cami. Another important term is the maximal gap. For two prime gaps Gi and Gj , i, j [1, ), Gj is a maximal gap if ∈ ∞ Gi <Gj i < j. Using exhaustive search, the largest known maximal prime gap having∀ length 1550 was found by Bertil Nyman[10] and occurs after the prime number 18361375334787046697. Bertrand’s Postulate states that for every integer k (k > 1), there exists a prime number P , such that, k<P< 2k. th Another form of the postulate is that, for the i prime, Pi, Pi+1 < 2Pi. (2) This implies that Gi <Pi. It was proposed as a conjecture initially, and proved by Chebyshev. This postulate is definitely weaker than the Prime k Number Theorem, since there are roughly primes upto k. This count is ln k greater than that stated by Bertrand’s postulate. An upper bound can be inferred on the gaps between primes using the Prime Number Theorem, as follows: For every ǫ> 0, there is a number N such that n>N, Gi <Piǫ. Also, ∀ G lim i =0 →∞ n Pi because the gaps get arbitrarily small compared to the prime numbers. On the other hand, Legendre’s Conjecture states that for every integer k,k > 1, there exists a prime number P , such that, k2 <P < (k + 1)2 . Cramer’s Conjecture This conjecture, devised by Harald Cramer in 1936[4] provides an asymptotic bound on prime gaps: 2 Pi+1 Pi = O (log Pi) (3) − where O represents the Big-Oh notation. There are several versions of Cramer’s Conjecture, of which (3) has not been proven yet. There are also weaker versions of this conjecture, which have been verified using conditional proofs, listed below: 4 1. Pi+1 Pi = O √Pi log Pi , which Cramer proved assuming the Reimann − Hypothesis[4]. 0.525 2. Pi+1 Pi = O (P ) proved by Baker, Harman and Pintz[3]. − i 3. With reference to the previous two points, prime gaps can grow more than logarithmically, therefore, the upper bound would have to be fur- ther worked on. It was proved that Pi+1 Pi lim sup − = . n→∞ log Pi ∞ 4. Point 3 was improved as under: 2 Pi+1 Pi (log log log Pi) lim sup − > 0. n→∞ log Pi · log log Pi log log log log Pi It was conjectured by Paul Erdos that the left hand side of the above equation is infinite, and this was successfully proven in [5]. A strong form of Cramer’s conjecture was provided by Daniel Shanks in [12] as 2 Gi log Pi ≈ th for the i prime Pi. The maximal gap Gi has been expressed in terms of the prime counting function as elaborated by Wolf in [14], Pi Gi (2 log π (Pi) log (Pi)+ c) ≈ π (Pi) − where c = log C2 =0.2778769... and C2 =1.3203236 is double the twin primes constant. Gauss’s approximation stated earlier, N πN , ≈ log N would reduce the previous equation to Gi log (Pi) (log (Pi) 2 log log (Pi)) , ≈ − 2 which for large values of Pi reduces to Cramer’s Conjecture, Gi log Pi. ≈ Zhang[15] proved that there are an infinite number of primes which have a prime gap less than 70,000(Eq. 1). After this, James Maynard reduced the bound [8] by proving that lim (Pn+1 Pn) 600. n→∞ − ≤ 5 The Polymath Project brought mathematicians to work together on extend- ing Zhang’s proof, and finally, the bound has been improvised and brought down to 246.
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