Using Formal Concept Analysis in Mathematical Discovery
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Pseudoprime Reductions of Elliptic Curves
Mathematical Proceedings of the Cambridge Philosophical Society VOL. 146 MAY 2009 PART 3 Math. Proc. Camb. Phil. Soc. (2009), 146, 513 c 2008 Cambridge Philosophical Society 513 doi:10.1017/S0305004108001758 Printed in the United Kingdom First published online 14 July 2008 Pseudoprime reductions of elliptic curves BY ALINA CARMEN COJOCARU Dept. of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, U.S.A. and The Institute of Mathematics of the Romanian Academy, Bucharest, Romania. e-mail: [email protected] FLORIAN LUCA Instituto de Matematicas,´ Universidad Nacional Autonoma´ de Mexico,´ C.P. 58089, Morelia, Michoacan,´ Mexico.´ e-mail: [email protected] AND IGOR E. SHPARLINSKI Dept. of Computing, Macquarie University, Sydney, NSW 2109, Australia. e-mail: [email protected] (Received 10 April 2007; revised 25 April 2008) Abstract Let b 2 be an integer and let E/Q be a fixed elliptic curve. In this paper, we estimate the number of primes p x such that the number of points n E (p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of n E (p). 1. Introduction Let b 2 be an integer. Recall that a pseudoprime to base b is a composite positive integer m such that the congruence bm ≡ b (mod m) holds. The question of the distri- bution of pseudoprimes in certain sequences of positive integers has received some in- terest. For example, in [PoRo], van der Poorten and Rotkiewicz show that any arithmetic 514 A. -
Recent Progress in Additive Prime Number Theory
Introduction Arithmetic progressions Other linear patterns Recent progress in additive prime number theory Terence Tao University of California, Los Angeles Mahler Lecture Series Terence Tao Recent progress in additive prime number theory Introduction Arithmetic progressions Other linear patterns Additive prime number theory Additive prime number theory is the study of additive patterns in the prime numbers 2; 3; 5; 7;:::. Examples of additive patterns include twins p; p + 2, arithmetic progressions a; a + r;:::; a + (k − 1)r, and prime gaps pn+1 − pn. Many open problems regarding these patterns still remain, but there has been some recent progress in some directions. Terence Tao Recent progress in additive prime number theory Random models for the primes Introduction Sieve theory Arithmetic progressions Szemerédi’s theorem Other linear patterns Putting it together Long arithmetic progressions in the primes I’ll first discuss a theorem of Ben Green and myself from 2004: Theorem: The primes contain arbitrarily long arithmetic progressions. Terence Tao Recent progress in additive prime number theory Random models for the primes Introduction Sieve theory Arithmetic progressions Szemerédi’s theorem Other linear patterns Putting it together It was previously established by van der Corput (1929) that the primes contained infinitely many progressions of length three. In 1981, Heath-Brown showed that there are infinitely many progressions of length four, in which three elements are prime and the fourth is an almost prime (the product of at most -
New Formulas for Semi-Primes. Testing, Counting and Identification
New Formulas for Semi-Primes. Testing, Counting and Identification of the nth and next Semi-Primes Issam Kaddouraa, Samih Abdul-Nabib, Khadija Al-Akhrassa aDepartment of Mathematics, school of arts and sciences bDepartment of computers and communications engineering, Lebanese International University, Beirut, Lebanon Abstract In this paper we give a new semiprimality test and we construct a new formula for π(2)(N), the function that counts the number of semiprimes not exceeding a given number N. We also present new formulas to identify the nth semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots 3 of N : P , P ....P 3 √N. 1 2 π( √N) ≤ Keywords: prime, semiprime, nth semiprime, next semiprime 1. Introduction Securing data remains a concern for every individual and every organiza- tion on the globe. In telecommunication, cryptography is one of the studies that permits the secure transfer of information [1] over the Internet. Prime numbers have special properties that make them of fundamental importance in cryptography. The core of the Internet security is based on protocols, such arXiv:1608.05405v1 [math.NT] 17 Aug 2016 as SSL and TSL [2] released in 1994 and persist as the basis for securing dif- ferent aspects of today’s Internet [3]. The Rivest-Shamir-Adleman encryption method [4], released in 1978, uses asymmetric keys for exchanging data. A secret key Sk and a public key Pk are generated by the recipient with the following property: A message enciphered Email addresses: [email protected] (Issam Kaddoura), [email protected] (Samih Abdul-Nabi) 1 by Pk can only be deciphered by Sk and vice versa. -
Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6. -
Hidden Multiscale Order in the Primes
HIDDEN MULTISCALE ORDER IN THE PRIMES SALVATORE TORQUATO, GE ZHANG, AND MATTHEW DE COURCY-IRELAND Abstract. We study the pair correlations between prime numbers in an in- terval M ≤ p ≤ M + L with M → ∞, L/M → β > 0. By analyzing the structure factor, we prove, conditionally on the Hardy-Littlewood conjecture on prime pairs, that the primes are characterized by unanticipated multiscale order. Specifically, their limiting structure factor is that of a union of an in- finite number of periodic systems and is characterized by dense set of Dirac delta functions. Primes in dyadic intervals are the first examples of what we call effectively limit-periodic point configurations. This behavior implies anomalously suppressed density fluctuations compared to uncorrelated (Pois- son) systems at large length scales, which is now known as hyperuniformity. Using a scalar order metric τ calculated from the structure factor, we iden- tify a transition between the order exhibited when L is comparable to M and the uncorrelated behavior when L is only logarithmic in M. Our analysis for the structure factor leads to an algorithm to reconstruct primes in a dyadic interval with high accuracy. 1. Introduction While prime numbers are deterministic, by some probabilistic descriptors, they can be regarded as pseudo-random in nature. Indeed, the primes can be difficult to distinguish from a random configuration of the same density. For example, assuming a plausible conjecture, Gallagher proved that the gaps between primes follow a Poisson distribution [13]. Thus the number of M X such that there are exactly N primes in the interval M<p<M + L of length≤ L λ ln X is given asymptotically by ∼ e λλN # M X such that π(M + L) π(M)= N X − . -
From Apollonian Circle Packings to Fibonacci Numbers
From Apollonian Circle Packings to Fibonacci Numbers Je↵Lagarias, University of Michigan March 25, 2009 2 Credits Results on integer Apollonian packings are joint work with • Ron Graham, Colin Mallows, Allan Wilks, Catherine Yan, ([GLMWY]) Some of the work on Fibonacci numbers is an ongoing joint • project with Jon Bober.([BL]) Work of J. L. was partially supported by NSF grant • DMS-0801029. 3 Table of Contents 1. Exordium 2. Apollonian Circle Packings 3. Fibonacci Numbers 4. Peroratio 4 1. Exordium (Contents of Talk)-1 The talk first discusses Apollonian circle packings. Then it • discusses integral Apollonian packings - those with all circles of integer curvature. These integers are describable in terms of integer orbits of • a group of 4 4 integer matrices of determinant 1, the A ⇥ ± Apollonian group, which is of infinite index in O(3, 1, Z), an arithmetic group acting on Lorentzian space. [Technically sits inside an integer group conjugate to O(3, 1, Z).] A 5 Contents of Talk-2 Much information on primality and factorization theory of • integers in such orbits can be read o↵using a sieve method recently developed by Bourgain, Gamburd and Sarnak. They observe: The spectral geometry of the Apollonian • group controls the number theory of such integers. One notable result: integer orbits contain infinitely many • almost prime vectors. 6 Contents of Talk-3 The talk next considers Fibonacci numbers and related • quantities. These can be obtained an orbit of an integer subgroup of 2 2 matrices of determinant 1, the F ⇥ ± Fibonacci group. This group is of infinite index in GL(2, Z), an arithmetic group acting on the upper and lower half planes. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Std XII Computer Science
J.H. TARAPORE SCHOOL WORKSHEET NO- 5 STD XII :A/B/C COMPUTER SCIENCE (2020-2021) Topic : Java Practical Programs General Instructions: • Give the program name as “yournameQ1” eg : “SamikshaQ1,Samiksha Q2” • Execute the programs in your laptop. • Save the program and output together in a Microsoft Word File. • Submit the pendrive on the first day of the school reopening. Question 1. Write a program to check whether the given number is a Prime Adam Number or Not. Prime Number:A number which has only two factors,i.e 1 and the number itself. Adam Number : The square of a number and the square of its reverse are reverse to each other.Example,if n=31 and reverse of n=13,then 312=961 and 132= 169,which is reverse of 961.Thus ,31 is an Adam Number. So, 31 is a Prime Adam Number. Question 2. Write a program to check whether the given number is a Evil Number or Not. An Evil number is a positive whole number which has even number of 1’s in its binary equivalent. Example: Binary equivalent of 9 is 1001, which contains even number of 1’s. Thus, 9 is an Evil Number. A few Evil numbers are 3, 5, 6, 9.... Question 3. Write a program to check whether the given number is a Smith Number or Not. A Smith Number is a composite number,the sum of the digits of a number is equal to the sum of the digits of its prime factors (excluding 1). Example: 666 Prime factors of 666 : 2,3,3 and 37. -
Which Polynomials Represent Infinitely Many Primes?
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 1 (2018), pp. 161-180 © Research India Publications http://www.ripublication.com/gjpam.htm Which polynomials represent infinitely many primes? Feng Sui Liu Department of Mathematics, NanChang University, NanChang, China. Abstract In this paper a new arithmetical model has been found by extending both basic operations + and × into finite sets of natural numbers. In this model we invent a recursive algorithm on sets of natural numbers to reformulate Eratosthene’s sieve. By this algorithm we obtain a recursive sequence of sets, which converges to the set of all natural numbers x such that x2 + 1 is prime. The corresponding cardinal sequence is strictly increasing. Test that the cardinal function is continuous with respect to an order topology, we immediately prove that there are infinitely many natural numbers x such that x2 + 1 is prime. Based on the reasoning paradigm above we further prove a general result: if an integer valued polynomial of degree k represents at least k +1 positive primes, then this polynomial represents infinitely many primes. AMS subject classification: Primary 11N32; Secondary 11N35, 11U09,11Y16, 11B37. Keywords: primes in polynomial, twin primes, arithmetical model, recursive sieve method, limit of sequence of sets, Ross-Littwood paradox, parity problem. 1. Introduction Whether an integer valued polynomial represents infinitely many primes or not, this is an intriguing question. Except some obvious exceptions one conjectured that the answer is yes. Example 1.1. Bouniakowsky [2], Schinzel [35], Bateman and Horn [1]. In this paper a recursive algorithm or sieve method adds some exotic structures to the sets of natural numbers, which allow us to answer the question: which polynomials represent infinitely many primes? 2 Feng Sui Liu In 1837, G.L. -
A Set of Sequences in Number Theory
A SET OF SEQUENCES IN NUMBER THEORY by Florentin Smarandache University of New Mexico Gallup, NM 87301, USA Abstract: New sequences are introduced in number theory, and for each one a general question: how many primes each sequence has. Keywords: sequence, symmetry, consecutive, prime, representation of numbers. 1991 MSC: 11A67 Introduction. 74 new integer sequences are defined below, followed by references and some open questions. 1. Consecutive sequence: 1,12,123,1234,12345,123456,1234567,12345678,123456789, 12345678910,1234567891011,123456789101112, 12345678910111213,... How many primes are there among these numbers? In a general form, the Consecutive Sequence is considered in an arbitrary numeration base B. Reference: a) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache. 2. Circular sequence: 1,12,21,123,231,312,1234,2341,3412,4123,12345,23451,34512,45123,51234, | | | | | | | | | --- --------- ----------------- --------------------------- 1 2 3 4 5 123456,234561,345612,456123,561234,612345,1234567,2345671,3456712,... | | | --------------------------------------- ---------------------- ... 6 7 How many primes are there among these numbers? 3. Symmetric sequence: 1,11,121,1221,12321,123321,1234321,12344321,123454321, 1234554321,12345654321,123456654321,1234567654321, 12345677654321,123456787654321,1234567887654321, 12345678987654321,123456789987654321,12345678910987654321, 1234567891010987654321,123456789101110987654321, 12345678910111110987654321,... How many primes are there among these numbers? In a general form, the Symmetric Sequence is considered in an arbitrary numeration base B. References: a) Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ 85287- 1006, USA. b) Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache. 4. Deconstructive sequence: 1,23,456,7891,23456,789123,4567891,23456789,123456789,1234567891, .. -
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. -
Eureka Issue 61
Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42).