Predicting Maximal Gaps in Sets of Primes
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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. -
Epiglass ® and Other International Paint Products
Epiglass® Multipurpose Epoxy Resin ABOUT THE AUTHORS ROGER MARSHALL For eight years award winning author Roger Marshall has been the technical editor for Soundings magazine where his articles are read by about 250,000 people each month. Marshall’s experience as a writer spans for many years. His work has appeared worldwide as well as the New York Times, Daily Telegraph (UK), Sports illustrated, Sail, Cruising World, Motor Boating and sailing, Yachting and many other newspapers and magazines. Marshall is also the author of twelve marine related books, two of which were translated into Italian and Spanish. His last book All about Powerboats was published by International Marine in the spring of 2002. He has another book Rough Weather Seamanship due in the fall of 2003 and is currently working on a new book, Elements of Powerboat Design for International Marine. But writing is only a small part of Marshall’s talents. He is also a designer of boats, both power and sail. After completing a program in small craft design at Southampton College in England, Marshall, who still holds a British passport, moved to the United States in 1973 to take a position at Sparkman & Stephens, Inc. in New York. He worked there as a designer for nearly 5 years and then left to establish his own yacht design studio in Jamestown, Rhode Island. As an independent designer he has designed a wide range of boats and was project engineer for the Courageous Challenge for the 1987 America’s Cup campaign in Australia. In 1999 one of his cruising yacht designs was selected for inclusion in Ocean Cruising magazine’s American Yacht Review. -
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. -
21 Pintura.Pdf
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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. -
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. -
Elementary Number Theory
Elementary Number Theory Peter Hackman HHH Productions November 5, 2007 ii c P Hackman, 2007. Contents Preface ix A Divisibility, Unique Factorization 1 A.I The gcd and B´ezout . 1 A.II Two Divisibility Theorems . 6 A.III Unique Factorization . 8 A.IV Residue Classes, Congruences . 11 A.V Order, Little Fermat, Euler . 20 A.VI A Brief Account of RSA . 32 B Congruences. The CRT. 35 B.I The Chinese Remainder Theorem . 35 B.II Euler’s Phi Function Revisited . 42 * B.III General CRT . 46 B.IV Application to Algebraic Congruences . 51 B.V Linear Congruences . 52 B.VI Congruences Modulo a Prime . 54 B.VII Modulo a Prime Power . 58 C Primitive Roots 67 iii iv CONTENTS C.I False Cases Excluded . 67 C.II Primitive Roots Modulo a Prime . 70 C.III Binomial Congruences . 73 C.IV Prime Powers . 78 C.V The Carmichael Exponent . 85 * C.VI Pseudorandom Sequences . 89 C.VII Discrete Logarithms . 91 * C.VIII Computing Discrete Logarithms . 92 D Quadratic Reciprocity 103 D.I The Legendre Symbol . 103 D.II The Jacobi Symbol . 114 D.III A Cryptographic Application . 119 D.IV Gauß’ Lemma . 119 D.V The “Rectangle Proof” . 123 D.VI Gerstenhaber’s Proof . 125 * D.VII Zolotareff’s Proof . 127 E Some Diophantine Problems 139 E.I Primes as Sums of Squares . 139 E.II Composite Numbers . 146 E.III Another Diophantine Problem . 152 E.IV Modular Square Roots . 156 E.V Applications . 161 F Multiplicative Functions 163 F.I Definitions and Examples . 163 CONTENTS v F.II The Dirichlet Product . -
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. -
Simple High-Level Code for Cryptographic Arithmetic - with Proofs, Without Compromises
Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Erbsen, Andres et al. “Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises.” Proceedings - IEEE Symposium on Security and Privacy, May-2019 (May 2019) © 2019 The Author(s) As Published 10.1109/SP.2019.00005 Publisher Institute of Electrical and Electronics Engineers (IEEE) Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/130000 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Simple High-Level Code For Cryptographic Arithmetic – With Proofs, Without Compromises Andres Erbsen Jade Philipoom Jason Gross Robert Sloan Adam Chlipala MIT CSAIL, Cambridge, MA, USA fandreser, jadep, [email protected], [email protected], [email protected] Abstract—We introduce a new approach for implementing where X25519 was the only arithmetic-based crypto primitive cryptographic arithmetic in short high-level code with machine- we need, now would be the time to declare victory and go checked proofs of functional correctness. We further demonstrate home. Yet most of the Internet still uses P-256, and the that simple partial evaluation is sufficient to transform such initial code into the fastest-known C code, breaking the decades- current proposals for post-quantum cryptosystems are far from old pattern that the only fast implementations are those whose Curve25519’s combination of performance and simplicity. instruction-level steps were written out by hand. -
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. -
Introduction to Abstract Algebra “Rings First”
Introduction to Abstract Algebra \Rings First" Bruno Benedetti University of Miami January 2020 Abstract The main purpose of these notes is to understand what Z; Q; R; C are, as well as their polynomial rings. Contents 0 Preliminaries 4 0.1 Injective and Surjective Functions..........................4 0.2 Natural numbers, induction, and Euclid's theorem.................6 0.3 The Euclidean Algorithm and Diophantine Equations............... 12 0.4 From Z and Q to R: The need for geometry..................... 18 0.5 Modular Arithmetics and Divisibility Criteria.................... 23 0.6 *Fermat's little theorem and decimal representation................ 28 0.7 Exercises........................................ 31 1 C-Rings, Fields and Domains 33 1.1 Invertible elements and Fields............................. 34 1.2 Zerodivisors and Domains............................... 36 1.3 Nilpotent elements and reduced C-rings....................... 39 1.4 *Gaussian Integers................................... 39 1.5 Exercises........................................ 41 2 Polynomials 43 2.1 Degree of a polynomial................................. 44 2.2 Euclidean division................................... 46 2.3 Complex conjugation.................................. 50 2.4 Symmetric Polynomials................................ 52 2.5 Exercises........................................ 56 3 Subrings, Homomorphisms, Ideals 57 3.1 Subrings......................................... 57 3.2 Homomorphisms.................................... 58 3.3 Ideals.........................................