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Quadratic Reciprocity and Computing Modular Square Roots.Pdf
12 Quadratic reciprocity and computing modular square roots In §2.8, we initiated an investigation of quadratic residues. This chapter continues this investigation. Recall that an integer a is called a quadratic residue modulo a positive integer n if gcd(a, n) = 1 and a ≡ b2 (mod n) for some integer b. First, we derive the famous law of quadratic reciprocity. This law, while his- torically important for reasons of pure mathematical interest, also has important computational applications, including a fast algorithm for testing if an integer is a quadratic residue modulo a prime. Second, we investigate the problem of computing modular square roots: given a quadratic residue a modulo n, compute an integer b such that a ≡ b2 (mod n). As we will see, there are efficient probabilistic algorithms for this problem when n is prime, and more generally, when the factorization of n into primes is known. 12.1 The Legendre symbol For an odd prime p and an integer a with gcd(a, p) = 1, the Legendre symbol (a j p) is defined to be 1 if a is a quadratic residue modulo p, and −1 otherwise. For completeness, one defines (a j p) = 0 if p j a. The following theorem summarizes the essential properties of the Legendre symbol. Theorem 12.1. Let p be an odd prime, and let a, b 2 Z. Then we have: (i) (a j p) ≡ a(p−1)=2 (mod p); in particular, (−1 j p) = (−1)(p−1)=2; (ii) (a j p)(b j p) = (ab j p); (iii) a ≡ b (mod p) implies (a j p) = (b j p); 2 (iv) (2 j p) = (−1)(p −1)=8; p−1 q−1 (v) if q is an odd prime, then (p j q) = (−1) 2 2 (q j p). -
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. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
The Circle Method Applied to Goldbach's Weak Conjecture
CARLETON UNIVERSITY SCHOOL OF MATHEMATICS AND STATISTICS HONOURS PROJECT TITLE: The Circle Method Applied to Goldbach’s Weak Conjecture AUTHOR: Michael Cayanan SUPERVISOR: Brandon Fodden DATE: August 27, 2019 Acknowledgements First and foremost, I would like to thank my supervisor Brandon Fodden not only for all of his advice and guidance throughout the writing of this paper, but also for being such a welcoming and understanding instructor during my first year of undergraduate study. I would also like to thank the second reader Kenneth Williams for graciously providing his time and personal insight into this paper. Of course, I am also very grateful for the countless professors and instructors who have offered me outlets to explore my curiosity and to cultivate my appreciation of mathematical abstraction through the courses that they have organized. I am eternally indebted to the love and continual support provided by my family and my dearest friends throughout my undergraduate journey. They were always able to provide assurance and uplift me during my worst and darkest moments, and words cannot describe how grateful I am to be surrounded with so much compassion and inspiration. This work is as much yours as it is mine, and I love you all 3000. THE CIRCLE METHOD APPLIED TO GOLDBACH'S WEAK CONJECTURE Michael Cayanan August 27, 2019 Abstract. Additive number theory is the branch of number theory that is devoted to the study of repre- sentations of integers subject to various arithmetic constraints. One of, if not, the most popular question that additive number theory asks arose in a letter written by Christian Goldbach to Leonhard Euler, fa- mously dubbed the Goldbach conjecture. -
On Sufficient Conditions for the Existence of Twin Values in Sieves
On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers by Luke Szramowski Submitted in Partial Fulfillment of the Requirements For the Degree of Masters of Science in the Mathematics Program Youngstown State University May, 2020 On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers Luke Szramowski I hereby release this thesis to the public. I understand that this thesis will be made available from the OhioLINK ETD Center and the Maag Library Circulation Desk for public access. I also authorize the University or other individuals to make copies of this thesis as needed for scholarly research. Signature: Luke Szramowski, Student Date Approvals: Dr. Eric Wingler, Thesis Advisor Date Dr. Thomas Wakefield, Committee Member Date Dr. Thomas Madsen, Committee Member Date Dr. Salvador A. Sanders, Dean of Graduate Studies Date Abstract For many years, a major question in sieve theory has been determining whether or not a sieve produces infinitely many values which are exactly two apart. In this paper, we will discuss a new result in sieve theory, which will give sufficient conditions for the existence of values which are exactly two apart. We will also show a direct application of this theorem on an existing sieve as well as detailing attempts to apply the theorem to the Sieve of Eratosthenes. iii Contents 1 Introduction 1 2 Preliminary Material 1 3 Sieves 5 3.1 The Sieve of Eratosthenes . 5 3.2 The Block Sieve . 9 3.3 The Finite Block Sieve . 12 3.4 The Sieve of Joseph Flavius . -
Generalized Perfect Numbers
Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 73–82 Generalized perfect numbers Antal Bege Kinga Fogarasi Sapientia–Hungarian University of Sapientia–Hungarian University of Transilvania Transilvania Department of Mathematics and Department of Mathematics and Informatics, Informatics, Tˆargu Mure¸s, Romania Tˆargu Mure¸s, Romania email: [email protected] email: [email protected] Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. A natural number is perfect if σ(n) = 2n. This concept was already generalized in form of superperfect numbers σ2(n)= σ(σ(n)) = k+1 k−1 2n and hyperperfect numbers σ(n)= k n + k . In this paper some new ways of generalizing perfect numbers are inves- tigated, numerical results are presented and some conjectures are estab- lished. 1 Introduction For the natural number n we denote the sum of positive divisors by arXiv:1008.0155v1 [math.NT] 1 Aug 2010 σ(n)= X d. d|n Definition 1 A positive integer n is called perfect number if it is equal to the sum of its proper divisors. Equivalently: σ(n)= 2n, where AMS 2000 subject classifications: 11A25, 11Y70 Key words and phrases: perfect number, superperfect number, k-hyperperfect number 73 74 A. Bege, K. Fogarasi Example 1 The first few perfect numbers are: 6, 28, 496, 8128, . (Sloane’s A000396 [15]), since 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Euclid discovered that the first four perfect numbers are generated by the for- mula 2n−1(2n − 1). -
Cubic and Biquadratic Reciprocity
Rational (!) Cubic and Biquadratic Reciprocity Paul Pollack 2005 Ross Summer Mathematics Program It is ordinary rational arithmetic which attracts the ordinary man ... G.H. Hardy, An Introduction to the Theory of Numbers, Bulletin of the AMS 35, 1929 1 Quadratic Reciprocity Law (Gauss). If p and q are distinct odd primes, then q p 1 q 1 p = ( 1) −2 −2 . p − q We also have the supplementary laws: 1 (p 1)/2 − = ( 1) − , p − 2 (p2 1)/8 and = ( 1) − . p − These laws enable us to completely character- ize the primes p for which a given prime q is a square. Question: Can we characterize the primes p for which a given prime q is a cube? a fourth power? We will focus on cubes in this talk. 2 QR in Action: From the supplementary law we know that 2 is a square modulo an odd prime p if and only if p 1 (mod 8). ≡± Or take q = 11. We have 11 = p for p 1 p 11 ≡ (mod 4), and 11 = p for p 1 (mod 4). p − 11 6≡ So solve the system of congruences p 1 (mod 4), p (mod 11). ≡ ≡ OR p 1 (mod 4), p (mod 11). ≡− 6≡ Computing which nonzero elements mod p are squares and nonsquares, we find that 11 is a square modulo a prime p = 2, 11 if and only if 6 p 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44). ≡ q Observe that the p with p = 1 are exactly the primes in certain arithmetic progressions. -
How Numbers Work Discover the Strange and Beautiful World of Mathematics
How Numbers Work Discover the strange and beautiful world of mathematics NEW SCIENTIST 629745_How_Number_Work_Book.indb 3 08/12/17 4:11 PM First published in Great Britain in 2018 by John Murray Learning First published in the US in 2018 by Nicholas Brealey Publishing John Murray Learning and Nicholas Brealey Publishing are companies of Hachette UK Copyright © New Scientist 2018 The right of New Scientist to be identified as the Author of the Work has been asserted by it in accordance with the Copyright, Designs and Patents Act 1988. Database right Hodder & Stoughton (makers) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, or as expressly permitted by law, or under terms agreed with the appropriate reprographic rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department at the addresses below. You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer. A catalogue record for this book is available from the British Library and the Library of Congress. UK ISBN 978 1 47 362974 5 / eISBN 978 14 7362975 2 US ISBN 978 1 47 367035 8 / eISBN 978 1 47367036 5 1 The publisher has used its best endeavours to ensure that any website addresses referred to in this book are correct and active at the time of going to press. -
Appendices A. Quadratic Reciprocity Via Gauss Sums
1 Appendices We collect some results that might be covered in a first course in algebraic number theory. A. Quadratic Reciprocity Via Gauss Sums A1. Introduction In this appendix, p is an odd prime unless otherwise specified. A quadratic equation 2 modulo p looks like ax + bx + c =0inFp. Multiplying by 4a, we have 2 2ax + b ≡ b2 − 4ac mod p Thus in studying quadratic equations mod p, it suffices to consider equations of the form x2 ≡ a mod p. If p|a we have the uninteresting equation x2 ≡ 0, hence x ≡ 0, mod p. Thus assume that p does not divide a. A2. Definition The Legendre symbol a χ(a)= p is given by 1ifa(p−1)/2 ≡ 1modp χ(a)= −1ifa(p−1)/2 ≡−1modp. If b = a(p−1)/2 then b2 = ap−1 ≡ 1modp,sob ≡±1modp and χ is well-defined. Thus χ(a) ≡ a(p−1)/2 mod p. A3. Theorem a The Legendre symbol ( p ) is 1 if and only if a is a quadratic residue (from now on abbre- viated QR) mod p. Proof.Ifa ≡ x2 mod p then a(p−1)/2 ≡ xp−1 ≡ 1modp. (Note that if p divides x then p divides a, a contradiction.) Conversely, suppose a(p−1)/2 ≡ 1modp.Ifg is a primitive root mod p, then a ≡ gr mod p for some r. Therefore a(p−1)/2 ≡ gr(p−1)/2 ≡ 1modp, so p − 1 divides r(p − 1)/2, hence r/2 is an integer. But then (gr/2)2 = gr ≡ a mod p, and a isaQRmodp. -
What Can Be Said About the Number 13 Beyond the Fact That It Is a Prime Number? *
CROATICA CHEMICA ACTA CCACAA 77 (3) 447¿456 (2004) ISSN-0011-1643 CCA-2946 Essay What Can Be Said about the Number 13 beyond the Fact that It Is a Prime Number? * Lionello Pogliani,a Milan Randi},b and Nenad Trinajsti}c,** aDipartimento di Chimica, Universitá della Calabria, 879030 Rende (CS), Italy bDepartment of Mathematics and Computer Science, Drake University, Des Moines, Iowa 50311, USA cThe Rugjer Bo{kovi} Institute, P.O. Box 180, HR-10002 Zagreb, Croatia RECEIVED DECEMBER 2, 2003; REVISED MARCH 8, 2004; ACCEPTED MARCH 9, 2004 Key words The story of the number 13 that goes back to ancient Egypt is told. The mathematical signifi- art cance of 13 is briefly reviewed and 13 is discussed in various contexts, with special reference chemistry to the belief of many that this number is a rather unlucky number. Contrary examples are also history presented. Regarding everything, the number 13 appears to be a number that leaves no one in- literature different. number theory poetry science INTRODUCTION – religious [3 in the Christian faith, 5 in Islam]; – dramatic [3 and 9 by William Shakespeare (1564– »Numero pondere et mensura Deus omnia condidit.« 1616) in Macbeth]; Isaac Newton in 1722 [ 1 – literary 3 and 10 by Graham Greene (1904–1991) (Rozsondai and Rozsondai) in the titles of his successful short novels The The concept of number is one of the oldest and most Third Man and The Tenth Man; 5 and 9 by Dorothy useful concepts in the history of human race.2–5 People Leigh Sayers (1893–1957) in the titles of her nov- have been aware of the number concept from the very els The Five Red Herrings and The Nine Taylors.