Large Gaps in Sets of Primes and Other Sequences I. Heuristics and Basic Constructions
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Asymptotic Formulae for the Number of Repeating Prime Sequences Less Than N
Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 22, 2016, No. 4, 29–40 Asymptotic formulae for the number of repeating prime sequences less than N Christopher L. Garvie Texas Natural Science Center, University of Texas at Austin 10100 Burnet Road, Austin, Texas 78758, USA e-mail: [email protected] Received: 21 April 2016 Accepted: 30 October 2016 Abstract: It is shown that prime sequences of arbitrary length, of which the prime pairs, (p; p+2), the prime triplet conjecture, (p; p+2; p+6) are simple examples, are true and that prime sequences of arbitrary length can be found and shown to repeat indefinitely. Asymptotic formulae compa- rable to the prime number theorem are derived for arbitrary length sequences. An elementary proof is also derived for the prime number theorem and Dirichlet’s Theorem on the arithmetic progression of primes. Keywords: Primes, Sequences, Distribution. AMS Classification: 11A41. 1 Introduction We build up the sequences of primes using a variant of the sieve process devised by Eratosthenes. In the sieve process the integers are written down sequentially up to the largest number we wish to check for primality. We first remove from consideration every number of the form 2n (n≥2), find the smallest number >2 that was not removed, that is 3 (the next prime), and similarly now remove all numbers of the form 3n (n≥2). As before find again the next number not removed, that is the next prime 5. The process is continued up to a desired prime Pr. -
Prime Numbers and Cryptography
the sunday tImes of malta maRCh 11, 2018 | 51 LIFE &WELLBEING SCIENCE Prime numbers and cryptography ALEXANDER FARRUGIA factored into its two primes by Benjamin Moody – that is a number having 155 digits! (By comparison, the number You are given a number n, and you 6,436,609 has only seven digits.) The first 50 prime numbers. are asked to find the two whole It took his desktop computer 73 numbers a and b, both greater days to achieve this feat. Later, than 1, such that a times b is equal during the same year, a 768-bit MYTH to n. If I give you the number 21, number, having 232 digits, was say, then you would tell me “that factored into its constituent DEBUNKED is 7 times 3”. Or, if I give you 55, primes by a team of 13 academ - you would say “that is 11 times 5”. ics – it took them two years to do Now suppose I ask you for the so! This is the largest number Is 1 a prime two whole numbers, both greater that has been factored to date. than 1, whose product (multipli - Nowadays, most websites number? cation) equals the number employing the RSA algorithm If this question was asked 6,436,609. This is a much harder use at least a 2,048-bit (617- before the start of the 20th problem, right? I invite you to try digit) number to encrypt data. century, one would have to work it out, before you read on. Some websites such as Facebook invariably received a ‘yes’. -
WXML Final Report: Prime Spacings
WXML Final Report: Prime Spacings Jayadev Athreya, Barry Forman, Kristin DeVleming, Astrid Berge, Sara Ahmed Winter 2017 1 Introduction 1.1 The initial problem In mathematics, prime gaps (defined as r = pn − pn−1) are widely studied due to the potential insights they may reveal about the distribution of the primes, as well as methods they could reveal about obtaining large prime numbers. Our research of prime spacings focused on studying prime intervals, which th is defined as pn − pn−1 − 1, where pn is the n prime. Specifically, we looked at what we defined as prime-prime intervals, that is primes having pn − pn−1 − 1 = r where r is prime. We have also researched how often different values of r show up as prime gaps or prime intervals, and why this distribution of values of r occurs the way that we have experimentally observed it to. 1.2 New directions The idea of studying the primality of prime intervals had not been previously researched, so there were many directions to take from the initial proposed problem. We saw multiple areas that needed to be studied: most notably, the prevalence of prime-prime intervals in the set of all prime intervals, and the distribution of the values of prime intervals. 1 2 Progress 2.1 Computational The first step taken was to see how many prime intervals up to the first nth prime were prime. Our original thoughts were that the amount of prime- prime intervals would grow at a rate of π(π(x)), since the function π tells us how many primes there are up to some number x. -
On the First Occurrences of Gaps Between Primes in a Residue Class
On the First Occurrences of Gaps Between Primes in a Residue Class Alexei Kourbatov JavaScripter.net Redmond, WA 98052 USA [email protected] Marek Wolf Faculty of Mathematics and Natural Sciences Cardinal Stefan Wyszy´nski University Warsaw, PL-01-938 Poland [email protected] To the memory of Professor Thomas R. Nicely (1943–2019) Abstract We study the first occurrences of gaps between primes in the arithmetic progression (P): r, r + q, r + 2q, r + 3q,..., where q and r are coprime integers, q > r 1. ≥ The growth trend and distribution of the first-occurrence gap sizes are similar to those arXiv:2002.02115v4 [math.NT] 20 Oct 2020 of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value dis- tribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are O(log2 x) first-occurrence gaps between primes in (P) below x, while the number of maximal gaps is only O(log x). We explore the con- nection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun’s constant. For the first occurrence of gap d in (P), we expect the end-of-gap prime p √d exp( d/ϕ(q)) infinitely often. Finally, we study the gap ≍ size as a function of its index in thep sequence of first-occurrence gaps. 1 1 Introduction Let pn be the n-th prime number, and consider the difference between successive primes, called a prime gap: pn+1 pn. -
Conjecture of Twin Primes (Still Unsolved Problem in Number Theory) an Expository Essay
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 12 (2017), 229 { 252 CONJECTURE OF TWIN PRIMES (STILL UNSOLVED PROBLEM IN NUMBER THEORY) AN EXPOSITORY ESSAY Hayat Rezgui Abstract. The purpose of this paper is to gather as much results of advances, recent and previous works as possible concerning the oldest outstanding still unsolved problem in Number Theory (and the most elusive open problem in prime numbers) called "Twin primes conjecture" (8th problem of David Hilbert, stated in 1900) which has eluded many gifted mathematicians. This conjecture has been circulating for decades, even with the progress of contemporary technology that puts the whole world within our reach. So, simple to state, yet so hard to prove. Basic Concepts, many and varied topics regarding the Twin prime conjecture will be cover. Petronas towers (Twin towers) Kuala Lumpur, Malaysia 2010 Mathematics Subject Classification: 11A41; 97Fxx; 11Yxx. Keywords: Twin primes; Brun's constant; Zhang's discovery; Polymath project. ****************************************************************************** http://www.utgjiu.ro/math/sma 230 H. Rezgui Contents 1 Introduction 230 2 History and some interesting deep results 231 2.1 Yitang Zhang's discovery (April 17, 2013)............... 236 2.2 "Polymath project"........................... 236 2.2.1 Computational successes (June 4, July 27, 2013)....... 237 2.2.2 Spectacular progress (November 19, 2013)........... 237 3 Some of largest (titanic & gigantic) known twin primes 238 4 Properties 240 5 First twin primes less than 3002 241 6 Rarefaction of twin prime numbers 244 7 Conclusion 246 1 Introduction The prime numbers's study is the foundation and basic part of the oldest branches of mathematics so called "Arithmetic" which supposes the establishment of theorems. -
Related Distribution of Prime Numbers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CSCanada.net: E-Journals (Canadian Academy of Oriental and... ISSN 1923-8444 [Print] Studies in Mathematical Sciences ISSN 1923-8452 [Online] Vol. 6, No. 2, 2013, pp. [96{103] www.cscanada.net DOI: 10.3968/j.sms.1923845220130602.478 www.cscanada.org Related Distribution of Prime Numbers Kaifu SONG[a],* [a] East Lake High School, Yichang, Hubei, China. * Corresponding author. Address: East Lake High School, Yichang 443100, Hubei, China; E-Mail: skf08@ sina.com Received: March 6, 2013/ Accepted: May 10, 2013/ Published: May 31, 2013 Abstract: The surplus model is established, and the distribution of prime numbers are solved by using the surplus model. Key words: Surplus model; Distribution of prime numbers Song, K. (2013). Related Distribution of Prime Numbers. Studies in Mathematical Sci- ences, 6 (2), 96{103. Available from http://www.cscanada.net/index.php/sms/article/ view/j.sms.1923845220130602.478 DOI: 10.3968/j.sms.1923845220130602.478 1. SURPLUS MODEL In the range of 1 ∼ N, r1 represents the quantity of numbers which do not contain 1 a factor 2, r = [N · ] ([x] is the largest integer not greater than x); r represents 1 2 2 2 the quantity of numbers which do not contain factor 3, r = [r · ]; r represents 2 1 3 3 4 the quantity of numbers do not contain factor 5, r = [r · ]; ... 3 2 5 In the range of 1 ∼ N, ri represents the quantity of numbers which do not contain factor pi , the method of the surplus numbers ri−1 to ri is called the residual model of related distribution of prime numbers. -
A New Approach to Prime Numbers A. INTRODUCTION
A new approach to prime numbers Pedro Caceres Longmeadow, MA- March 20th, 2017 Email: [email protected] Phone: +1 (763) 412-8915 Abstract: A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors. Primes can thus be considered the “basic building blocks”, the atoms, of the natural numbers. There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behavior of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n. The way to build the sequence of prime numbers uses sieves, an algorithm yielding all primes up to a given limit, using only trial division method which consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. -
Mathematical Constants and Sequences
Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) .. -
Article Title: the Nicholson's Conjecture Authors: Jan Feliksiak[1] Affiliations: N
Article title: The Nicholson's conjecture Authors: Jan Feliksiak[1] Affiliations: N. A.[1] Orcid ids: 0000-0002-9388-1470[1] Contact e-mail: [email protected] License information: This work has been published open access under Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at https://www.scienceopen.com/. Preprint statement: This article is a preprint and has not been peer-reviewed, under consideration and submitted to ScienceOpen Preprints for open peer review. DOI: 10.14293/S2199-1006.1.SOR-.PPT1JIB.v1 Preprint first posted online: 24 February 2021 Keywords: Distribution of primes, Prime gaps bound, Maximal prime gaps bound, Nicholson's conjecture, Prime number theorem THE NICHOLSON'S CONJECTURE JAN FELIKSIAK Abstract. This research paper discusses the distribution of the prime numbers, from the point of view of the Nicholson's Conjecture of 2013: ! n p(n+1) ≤ n log n p(n) The proof of the conjecture permits to develop and establish a Supremum bound on: ! n 1 p(n+1) log n − ≤ SUP n p(n) Nicholson's Conjecture belongs to the class of the strongest bounds on maximal prime gaps. 2000 Mathematics Subject Classification. 0102, 11A41, 11K65, 11L20, 11N05, 11N37,1102, 1103. Key words and phrases. Distribution of primes, maximal prime gaps upper bound, Nicholson's conjecture, Prime Number Theorem. 1 2 JAN FELIKSIAK 1. Preliminaries Within the scope of the paper, prime gap of the size g 2 N j g ≥ 2 is defined as an interval between two primes pn; p(n+1), containing (g − 1) composite integers. -
On the Infinity of Twin Primes and Other K-Tuples
On The Infinity of Twin Primes and other K-tuples Jabari Zakiya: [email protected] December 13, 2019 Abstract The paper uses the structure and math of Prime Generators to show there are an infinity of twin primes, proving the Twin Prime Conjecture, as well as establishing the infinity of other k-tuples of primes. 1 Introduction In number theory Polignac’s Conjecture (1849) [6] states there are infinitely many consecutive primes (prime pairs) that differ by any even number n. The Twin Prime Conjecture derives from it for prime pairs that differ by 2, the so called twin primes, e.g. (11, 13) and (101, 103). K-tuples are groupings of primes adhering to specific patterns, usually designated as (k, d) groupings, where k is the number of primes in the group and d the total spacing between its first and last prime [4]. Thus, Polignac’s pairs are type (2, n), where n is any even number, and twin primes are the specific named k-tuples of type (2, 2). Various types of k-tuples form a constellation of groupings for k ≥ 2. Triplets are type (3, 6) which have two forms, (0, 2, 6) and (0, 4, 6). The smallest occurrence for each form are (5, 7, 11) and (7, 11, 13). Some k-tuples have been given names. Three named (2, d) tuples are Twin Primes (2, 2), Cousin Primes (2, 4), and Sexy Primes (2, 6). The paper shows there are many more Sexy Primes than Twins or Cousins, though an infinity of each. The nature of the proof presented herein takes a totally different approach than usually taken. -
Generator and Verification Algorithms for Prime K−Tuples Using Binomial
International Mathematical Forum, Vol. 6, 2011, no. 44, 2169 - 2178 Generator and Verification Algorithms for Prime k−Tuples Using Binomial Expressions Zvi Retchkiman K¨onigsberg Instituto Polit´ecnico Nacional CIC Mineria 17-2, Col. Escandon Mexico D.F 11800, Mexico [email protected] Abstract In this paper generator and verification algorithms for prime k-tuples based on the the divisibility properties of binomial coefficients are intro- duced. The mathematical foundation lies in the connection that exists between binomial coefficients and the number of carries that result in the sum in different bases of the variables that form the binomial co- efficent and, characterizations of k-tuple primes in terms of binomial coefficients. Mathematics Subject Classification: 11A41, 11B65, 11Y05, 11Y11, 11Y16 Keywords: Prime k-tuples, Binomial coefficients, Algorithm 1 Introduction In this paper generator and verification algorithms fo prime k-tuples based on the divisibility properties of binomial coefficients are presented. Their math- ematical justification results from the work done by Kummer in 1852 [1] in relation to the connection that exists between binomial coefficients and the number of carries that result in the sum in different bases of the variables that form the binomial coefficient. The necessary and sufficient conditions provided for k-tuple primes verification in terms of binomial coefficients were inspired in the work presented in [2], however its proof is based on generating func- tions which is distinct to the argument provided here to prove it, for another characterization of this type see [3]. The mathematical approach applied to prove the presented results is novice, and the algorithms are new. -
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