Primality, Fractality, and Image Analysis
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An Amazing Prime Heuristic.Pdf
This document has been moved to https://arxiv.org/abs/2103.04483 Please use that version instead. AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 264955 1: · The very next day Giovanni La Barbera found the new record primes: 1693965 266443 1: · The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to N dx 2C2N 2C2 2 2 Z2 (log x) ∼ (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture. -
X9.80–2005 Prime Number Generation, Primality Testing, and Primality Certificates
This is a preview of "ANSI X9.80:2005". Click here to purchase the full version from the ANSI store. American National Standard for Financial Services X9.80–2005 Prime Number Generation, Primality Testing, and Primality Certificates Accredited Standards Committee X9, Incorporated Financial Industry Standards Date Approved: August 15, 2005 American National Standards Institute American National Standards, Technical Reports and Guides developed through the Accredited Standards Committee X9, Inc., are copyrighted. Copying these documents for personal or commercial use outside X9 membership agreements is prohibited without express written permission of the Accredited Standards Committee X9, Inc. For additional information please contact ASC X9, Inc., P.O. Box 4035, Annapolis, Maryland 21403. This is a preview of "ANSI X9.80:2005". Click here to purchase the full version from the ANSI store. ANS X9.80–2005 Foreword Approval of an American National Standard requires verification by ANSI that the requirements for due process, consensus, and other criteria for approval have been met by the standards developer. Consensus is established when, in the judgment of the ANSI Board of Standards Review, substantial agreement has been reached by directly and materially affected interests. Substantial agreement means much more than a simple majority, but not necessarily unanimity. Consensus requires that all views and objections be considered, and that a concerted effort be made toward their resolution. The use of American National Standards is completely voluntary; their existence does not in any respect preclude anyone, whether he has approved the standards or not from manufacturing, marketing, purchasing, or using products, processes, or procedures not conforming to the standards. -
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. -
A Short and Simple Proof of the Riemann's Hypothesis
A Short and Simple Proof of the Riemann’s Hypothesis Charaf Ech-Chatbi To cite this version: Charaf Ech-Chatbi. A Short and Simple Proof of the Riemann’s Hypothesis. 2021. hal-03091429v10 HAL Id: hal-03091429 https://hal.archives-ouvertes.fr/hal-03091429v10 Preprint submitted on 5 Mar 2021 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. A Short and Simple Proof of the Riemann’s Hypothesis Charaf ECH-CHATBI ∗ Sunday 21 February 2021 Abstract We present a short and simple proof of the Riemann’s Hypothesis (RH) where only undergraduate mathematics is needed. Keywords: Riemann Hypothesis; Zeta function; Prime Numbers; Millennium Problems. MSC2020 Classification: 11Mxx, 11-XX, 26-XX, 30-xx. 1 The Riemann Hypothesis 1.1 The importance of the Riemann Hypothesis The prime number theorem gives us the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper[1], it asserts that all the ’non-trivial’ zeros of the zeta function are complex numbers with real part 1/2. 1.2 Riemann Zeta Function For a complex number s where ℜ(s) > 1, the Zeta function is defined as the sum of the following series: +∞ 1 ζ(s)= (1) ns n=1 X In his 1859 paper[1], Riemann went further and extended the zeta function ζ(s), by analytical continuation, to an absolutely convergent function in the half plane ℜ(s) > 0, minus a simple pole at s = 1: s +∞ {x} ζ(s)= − s dx (2) s − 1 xs+1 Z1 ∗One Raffles Quay, North Tower Level 35. -
Bayesian Perspectives on Mathematical Practice
Bayesian perspectives on mathematical practice James Franklin University of New South Wales, Sydney, Australia In: B. Sriraman, ed, Handbook of the History and Philosophy of Mathematical Practice, Springer, 2020. Contents 1. Introduction 2. The relation of conjectures to proof 3. Applied mathematics and statistics: understanding the behavior of complex models 4. The objective Bayesian perspective on evidence 5. Evidence for and against the Riemann Hypothesis 6. Probabilistic relations between necessary truths? 7. The problem of induction in mathematics 8. Conclusion References Abstract Mathematicians often speak of conjectures as being confirmed by evi- dence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long re- sisted proof, such as the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in com- puter power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore be- come (among other things) an experimental science (though that has not dimin- ished the importance of proof in the traditional style). We examine how the evalu- ation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Nu- merical evidence in mathematics is related to the problem of induction; the occur- rence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical rea- soning. -
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Science Horizon Volume 6 Issue 4 April, 2021 President, Odisha Bigyan Academy Editorial Board Sri Umakant Swain Prof. Niranjan Barik Prof. Ramesh Chandra Parida Editor Er. Mayadhar Swain Dr. Choudhury Satyabrata Nanda Dr. Rajballav Mohanty Managing Editor Dr. Nilambar Biswal Er. Bhagat Charan Mohanty Language Expert Secretary, Odisha Bigyan Academy Prof. Sachidananda Tripathy HJCONTENTS HJ Subject Author Page 1. Editorial : Uttarakhand Glacial Burst-A Warning Er. Mayadhar Swain 178 2. The Star World Prof. B.B. Swain 180 3. Role of Physics in Information and Communication Dr. Sadasiva Biswal 186 Technology 4. National Mathematics Day, National Mathematics Binapani Saikrupa 188 Park & Srinivasa Ramanujan 5. A Brief History of Indian Math-Magicians (Part IV) Purusottam Sahoo 192 6. Glacier Dr. Manas Ranjan Senapati 197 7. Do Plants Recognise their Kins? Dr. Taranisen Panda 198 Dr. Raj Ballav Mohanty 8. Paris Climate Agreement and India’s Response Dr. Sundara Narayana Patro 201 9. Fluoride Contamination in Groundwater with Jayant Kumar Sahoo 205 Special Reference to Odisha 10. Electricity from Trees Dr. Ramesh Chandra Parida 209 11. Hydrogen as a Renewable Energy for Sustainable Prof. (Dr.) L.M. Das 212 Future 12. Nomophobia Dr. Namita Kumari Das 217 13. Quiz: Mammals Dr. Kedareswar Pradhan 220 14. Recent News on Science & Technology 222 The Cover Page depicts : Green Hydrogen Cover Design : Kalakar Sahoo APRIL, 2021 // EDITORIAL // UTTARAKHAND GLACIAL BURST- A WARNING Er. Mayadhar Swain On 7th February 2021, a flash flood occurred in Chamoli district of Uttarakhand villagers, mostly women, were uprooted in causing devastating damage and killing 204 this flood. At Reni, Risiganga joins the river persons. -
Ramanujan and Labos Primes, Their Generalizations, and Classifications
1 2 Journal of Integer Sequences, Vol. 15 (2012), 3 Article 12.1.1 47 6 23 11 Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes Vladimir Shevelev Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105 Israel [email protected] Abstract We study the parallel properties of the Ramanujan primes and a symmetric counter- part, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems. 1 Introduction The very well-known Bertrand postulate (1845) states that, for every x > 1, there exists a prime in the interval (x, 2x). This postulate quickly became a theorem when, in 1851, it was unexpectedly proved by Chebyshev (for a very elegant version of his proof, see Theorem 9.2 in [8]). In 1919, Ramanujan ([6]; [7, pp. 208–209]) gave a quite new and very simple proof of Bertrand’s postulate. Moreover, in his proof of a generalization of Bertrand’s, the following sequence of primes appeared ([11], A104272) 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167,... (1) Definition 1. For n ≥ 1, the n-th Ramanujan prime is the largest prime (Rn) for which π(Rn) − π(Rn/2) = n. Let us show that, equivalently, Rn is the smallest positive integer g(n) with the property that, if x ≥ g(n), then π(x) − π(x/2) ≥ n. -
Ramanujan Primes: Bounds, Runs, Twins, and Gaps
Ramanujan Primes: Bounds, Runs, Twins, and Gaps Jonathan Sondow 209 West 97th Street New York, NY 10025 USA [email protected] John W. Nicholson P. O. Box 2423 Arlington, TX 76004 USA [email protected] Tony D. Noe 14025 NW Harvest Lane Portland, OR 97229 USA [email protected] arXiv:1105.2249v2 [math.NT] 1 Aug 2011 Abstract The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, 1 then the interval 2 x,x contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below 10n of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. The Appendix explains Noe’s fast algorithm for computing R1,R2,...,Rn. 1 1 Introduction For n ≥ 1, the nth Ramanujan prime is defined as the smallest positive integer Rn with 1 the property that for any x ≥ Rn, there are at least n primes p with 2 x<p ≤ x. By its 1 minimality, Rn is indeed a prime, and the interval 2 Rn, Rn contains exactly n primes [10]. In 1919 Ramanujan proved a result which implies that Rn exists, and he gave the first five Ramanujan primes. -
2 Primes Numbers
V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 2 Primes Numbers Definition 2.1 A number is prime is it is greater than 1, and its only divisors are itself and 1. A number is called composite if it is greater than 1 and is the product of two numbers greater than 1. Thus, the positive numbers are divided into three mutually exclusive classes. The prime numbers, the composite numbers, and the unit 1. We can show that a number is composite numbers by finding a non-trivial factorization.8 Thus, 21 is composite because 21 = 3 7. The letter p is usually used to denote a prime number. × The first fifteen prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 These are found by a process of elimination. Starting with 2, 3, 4, 5, 6, 7, etc., we eliminate the composite numbers 4, 6, 8, 9, and so on. There is a systematic way of finding a table of all primes up to a fixed number. The method is called a sieve method and is called the Sieve of Eratosthenes9. We first illustrate in a simple example by finding all primes from 2 through 25. We start by listing the candidates: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 The first number 2 is a prime, but all multiples of 2 after that are not. Underline these multiples. They are eliminated as composite numbers. The resulting list is as follows: 2, 3, 4,5,6,7,8,9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,25 The next number not eliminated is 3, the next prime. -
Prime Number Theorem
Prime Number Theorem Bent E. Petersen Contents 1 Introduction 1 2Asymptotics 6 3 The Logarithmic Integral 9 4TheCebyˇˇ sev Functions θ(x) and ψ(x) 11 5M¨obius Inversion 14 6 The Tail of the Zeta Series 16 7 The Logarithm log ζ(s) 17 < s 8 The Zeta Function on e =1 21 9 Mellin Transforms 26 10 Sketches of the Proof of the PNT 29 10.1 Cebyˇˇ sev function method .................. 30 10.2 Modified Cebyˇˇ sev function method .............. 30 10.3 Still another Cebyˇˇ sev function method ............ 30 10.4 Yet another Cebyˇˇ sev function method ............ 31 10.5 Riemann’s method ...................... 31 10.6 Modified Riemann method .................. 32 10.7 Littlewood’s Method ..................... 32 10.8 Ikehara Tauberian Theorem ................. 32 11ProofofthePrimeNumberTheorem 33 References 39 1 Introduction Let π(x) be the number of primes p ≤ x. It was discovered empirically by Gauss about 1793 (letter to Enke in 1849, see Gauss [9], volume 2, page 444 and Goldstein [10]) and by Legendre (in 1798 according to [14]) that x π(x) ∼ . log x This statement is the prime number theorem. Actually Gauss used the equiva- lent formulation (see page 10) Z x dt π(x) ∼ . 2 log t 1 B. E. Petersen Prime Number Theorem For some discussion of Gauss’ work see Goldstein [10] and Zagier [45]. In 1850 Cebyˇˇ sev [3] proved a result far weaker than the prime number theorem — that for certain constants 0 <A1 < 1 <A2 π(x) A < <A . 1 x/log x 2 An elementary proof of Cebyˇˇ sev’s theorem is given in Andrews [1]. -
Notes on the Prime Number Theorem
Notes on the prime number theorem Kenji Kozai May 2, 2014 1 Statement We begin with a definition. Definition 1.1. We say that f(x) and g(x) are asymptotic as x → ∞, f(x) written f ∼ g, if limx→∞ g(x) = 1. The prime number theorem tells us about the asymptotic behavior of the number of primes that are less than a given number. Let π(x) be the number of primes numbers p such that p ≤ x. For example, π(2) = 1, π(3) = 2, π(4) = 2, etc. The statement that we will try to prove is as follows. Theorem 1.2 (Prime Number Theorem, p. 382). The asymptotic behavior of π(x) as x → ∞ is given by x π(x) ∼ . log x This was originally observed as early as the 1700s by Gauss and others. The first proof was given in 1896 by Hadamard and de la Vall´ee Poussin. We will give an overview of simplified proof. Most of the material comes from various sections of Gamelin – mostly XIV.1, XIV.3 and XIV.5. 2 The Gamma Function The gamma function Γ(z) is a meromorphic function that extends the fac- torial n! to arbitrary complex values. For Re z > 0, we can find the gamma function by the integral: ∞ Γ(z)= e−ttz−1dt, Re z > 0. Z0 1 To see the relation with the factorial, we integrate by parts: ∞ Γ(z +1) = e−ttzdt Z0 ∞ z −t ∞ −t z−1 = −t e |0 + z e t dt Z0 = zΓ(z). ∞ −t This holds whenever Re z > 0. -
Sequences of Numbers Obtained by Digit and Iterative Digit Sums of Sophie Germain Primes and Its Variants
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1473-1480 © Research India Publications http://www.ripublication.com Sequences of numbers obtained by digit and iterative digit sums of Sophie Germain primes and its variants 1Sheila A. Bishop, *1Hilary I. Okagbue, 2Muminu O. Adamu and 3Funminiyi A. Olajide 1Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria. 2Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria. 3Department of Computer and Information Sciences, Covenant University, Canaanland, Ota, Nigeria. Abstract Sequences were generated by the digit and iterative digit sums of Sophie Germain and Safe primes and their variants. The results of the digit and iterative digit sum of Sophie Germain and Safe primes were almost the same. The same applied to the square and cube of the respective primes. Also, the results of the digit and iterative digit sum of primes that are not Sophie Germain are the same with the primes that are notSafe. The results of the digit and iterative digit sum of prime that are either Sophie Germain or Safe are like the combination of the results of the respective primes when considered separately. Keywords: Sophie Germain prime, Safe prime, digit sum, iterative digit sum. Introduction In number theory, many types of primes exist; one of such is the Sophie Germain prime. A prime number p is said to be a Sophie Germain prime if 21p is also prime. A prime number qp21 is known as a safe prime. Sophie Germain prime were named after the famous French mathematician, physicist and philosopher Sophie Germain (1776-1831).