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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. -
An Analysis of Primality Testing and Its Use in Cryptographic Applications
An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability. -
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. -
MATH 404: ARITHMETIC 1. Divisibility and Factorization After Learning To
MATH 404: ARITHMETIC S. PAUL SMITH 1. Divisibility and Factorization After learning to count, add, and subtract, children learn to multiply and divide. The notion of division makes sense in any ring and much of the initial impetus for the development of abstract algebra arose from problems of division√ and factoriza- tion, especially in rings closely related to the integers such as Z[ d]. In the next several chapters we will follow this historical development by studying arithmetic in these and similar rings. √ To see that there are some serious issues to be addressed notice that in Z[ −5] the number 6 factorizes as a product of irreducibles in two different ways, namely √ √ 6 = 2.3 = (1 + −5)(1 − −5). 1 It isn’t hard to show these factors are irreducible but we postpone√ that for now . The key point for the moment is to note that this implies that Z[ −5] is not a unique factorization domain. This is in stark contrast to the integers. Since our teenage years we have taken for granted the fact that every integer can be written as a product of primes, which in Z are the same things as irreducibles, in a unique way. To emphasize the fact that there are questions to be understood, let us mention that Z[i] is a UFD. Even so, there is the question which primes in Z remain prime in Z[i]? This is a question with a long history: Fermat proved in ??? that a prime p in Z remains prime in Z[i] if and only if it is ≡ 3(mod 4).√ It is then natural to ask for each integer d, which primes in Z remain prime in Z[ d]? These and closely related questions lie behind much of the material we cover in the next 30 or so pages. -
The Prime Number Theorem a PRIMES Exposition
The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1. -
Ramanujan, Robin, Highly Composite Numbers, and the Riemann Hypothesis
Contemporary Mathematics Volume 627, 2014 http://dx.doi.org/10.1090/conm/627/12539 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis Jean-Louis Nicolas and Jonathan Sondow Abstract. We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical background of our subject. The second part is on its history, which includes several surprises. 1. Mathematical Background Definition. The sum-of-divisors function σ is defined by 1 σ(n):= d = n . d d|n d|n In 1913, Gr¨onwall found the maximal order of σ. Gr¨onwall’s Theorem [8]. The function σ(n) G(n):= (n>1) n log log n satisfies lim sup G(n)=eγ =1.78107 ... , n→∞ where 1 1 γ := lim 1+ + ···+ − log n =0.57721 ... n→∞ 2 n is the Euler-Mascheroni constant. Gr¨onwall’s proof uses: Mertens’s Theorem [10]. If p denotes a prime, then − 1 1 1 lim 1 − = eγ . x→∞ log x p p≤x 2010 Mathematics Subject Classification. Primary 01A60, 11M26, 11A25. Key words and phrases. Riemann Hypothesis, Ramanujan’s Theorem, Robin’s Theorem, sum-of-divisors function, highly composite number, superabundant, colossally abundant, Euler phi function. ©2014 American Mathematical Society 145 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 146 JEAN-LOUIS NICOLAS AND JONATHAN SONDOW Figure 1. Thomas Hakon GRONWALL¨ (1877–1932) Figure 2. Franz MERTENS (1840–1927) Nowwecometo: Ramanujan’s Theorem [2, 15, 16]. -
GRACEFUL and PRIME LABELINGS -Algorithms, Embeddings and Conjectures
GRACEFUL AND PRIME LABELINGS-Algorithms, Embeddings and Conjectures Group Discussion (KREC, Surathkal: Aug.16-25, 1999). Improved Version. GRACEFUL AND PRIME LABELINGS -Algorithms, Embeddings and Conjectures Suryaprakash Nagoji Rao* Exploration Business Group MRBC, ONGC, Mumbai-400 022 Abstract. Four algorithms giving rise to graceful graphs from a known (non-)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and highly graceful graphs. The Ringel-Rosa-Kotzig Conjecture is generalized to highly graceful graphs. Mayeda-Seshu Tree Generation Algorithm is modified to generate all possible graceful labelings of trees of order p. An alternative algorithm in terms of integers modulo p is described which includes all possible graceful labelings of trees of order p and some interesting properties are observed. Optimal and graceful graph embeddings (not necessarily connected) are given. Alternative proofs for embedding a graph into a graceful graph as a subgraph and as an induced subgraph are included. An algorithm to obtain an optimal graceful embedding is described. A necessary condition for a graph to be supergraceful is given. As a consequence some classes of non-supergraceful graphs are obtained. Embedding problems of a graph into a supergraceful graph are studied. A catalogue of super graceful graphs with at most five nodes is appended. Optimal graceful and supergraceful embeddings of a graph are given. Graph theoretical properties of prime and superprime graphs are listed. Good upper bound for minimum number of edges in a nonprime graph is given and some conjectures are proposed which in particular includes Entringer’s prime tree conjecture. -
ON the PRIMALITY of N! ± 1 and 2 × 3 × 5 ×···× P
MATHEMATICS OF COMPUTATION Volume 71, Number 237, Pages 441{448 S 0025-5718(01)01315-1 Article electronically published on May 11, 2001 ON THE PRIMALITY OF n! 1 AND 2 × 3 × 5 ×···×p 1 CHRIS K. CALDWELL AND YVES GALLOT Abstract. For each prime p,letp# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n! 1andp# 1 are known and have found two new primes of the first form (6380! + 1; 6917! − 1) and one of the second (42209# + 1). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found. 1. Introduction For each prime p,letp# be the product of the primes less than or equal to p. About 350 BC Euclid proved that there are infinitely many primes by first assuming they are only finitely many, say 2; 3;:::;p, and then considering the factorization of p#+1: Since then amateurs have expected many (if not all) of the values of p# 1 and n! 1 to be prime. Careful checks over the last half-century have turned up relatively few such primes [5, 7, 13, 14, 19, 25, 32, 33]. Using a program written by the second author, we greatly extended the previous search limits [8] from n ≤ 4580 for n! 1ton ≤ 10000, and from p ≤ 35000 for p# 1top ≤ 120000. This search took over a year of CPU time and has yielded three new primes: 6380!+1, 6917!−1 and 42209# + 1. -
Sum of Divisors of the Primorial and Sum of Squarefree Parts
International Mathematical Forum, Vol. 12, 2017, no. 7, 331 - 338 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7113 Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts Rafael Jakimczuk Divisi´onMatem´atica,Universidad Nacional de Luj´an Buenos Aires, Argentina Copyright c 2017 Rafael Jakimczuk. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Abstract Let pn be the n-th prime and σ(n) the sum of the positive divisors of n. Let us consider the primorial Pn = p1:p2 : : : pn, the sum of its Qn positive divisors is σ(Pn) = i=1(1 + pi). In the first section we prove the following asymptotic formula n 6 σ(P ) = Y(1 + p ) = eγ P log p + O (P ) : n i π2 n n n i=1 Let a(k) be the squarefree part of k, in the second section we prove the formula π2 X a(k) = x2 + o(x2): 30 1≤k≤x We also study integers with restricted squarefree parts and generalize these results to s-th free parts. Mathematics Subject Classification: 11A99, 11B99 Keywords: Primorial, divisors, squarefree parts, s-th free parts, average of arithmetical functions 332 Rafael Jakimczuk 1 Sum of Divisors of the Primorial In this section p denotes a positive prime and pn denotes the n-th prime. The following Mertens's formulae are well-known (see [5, Chapter VI]) 1 1 ! X = log log x + M + O ; (1) p≤x p log x where M is called Mertens's constant. -
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) .. -
Various Arithmetic Functions and Their Applications
University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2016 Various Arithmetic Functions and their Applications Florentin Smarandache University of New Mexico, [email protected] Octavian Cira Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Applied Mathematics Commons, Logic and Foundations Commons, Number Theory Commons, and the Set Theory Commons Recommended Citation Smarandache, Florentin and Octavian Cira. "Various Arithmetic Functions and their Applications." (2016). https://digitalrepository.unm.edu/math_fsp/256 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Octavian Cira Florentin Smarandache Octavian Cira and Florentin Smarandache Various Arithmetic Functions and their Applications Peer reviewers: Nassim Abbas, Youcef Chibani, Bilal Hadjadji and Zayen Azzouz Omar Communicating and Intelligent System Engineering Laboratory, Faculty of Electronics and Computer Science University of Science and Technology Houari Boumediene 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria Octavian Cira Florentin Smarandache Various Arithmetic Functions and their Applications PONS asbl Bruxelles, 2016 © 2016 Octavian Cira, Florentin Smarandache & Pons. All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners Pons asbl Quai du Batelage no. -
13 Lectures on Fermat's Last Theorem
I Paulo Ribenboim 13 Lectures on Fermat's Last Theorem Pierre de Fermat 1608-1665 Springer-Verlag New York Heidelberg Berlin Paulo Ribenboim Department of Mathematics and Statistics Jeffery Hall Queen's University Kingston Canada K7L 3N6 Hommage a AndrC Weil pour sa Leqon: goat, rigueur et pCnCtration. AMS Subiect Classifications (1980): 10-03, 12-03, 12Axx Library of Congress Cataloguing in Publication Data Ribenboim, Paulo. 13 lectures on Fermat's last theorem. Includes bibliographies and indexes. 1. Fermat's theorem. I. Title. QA244.R5 512'.74 79-14874 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. @ 1979 by Springer-Verlag New York Inc. Printed in the United States of America. 987654321 ISBN 0-387-90432-8 Springer-Verlag New York ISBN 3-540-90432-8 Springer-Verlag Berlin Heidelberg Preface Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians for more than three centuries. Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topic is presented in the form of lectures, where I survey the main lines of work on the problem. In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in suc- cession the main theories connected with the problem.