Generalised Mersenne Numbers Revisited
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1 Mersenne Primes and Perfect Numbers
1 Mersenne Primes and Perfect Numbers Basic idea: try to construct primes of the form an − 1; a, n ≥ 1. e.g., 21 − 1 = 3 but 24 − 1=3· 5 23 − 1=7 25 − 1=31 26 − 1=63=32 · 7 27 − 1 = 127 211 − 1 = 2047 = (23)(89) 213 − 1 = 8191 Lemma: xn − 1=(x − 1)(xn−1 + xn−2 + ···+ x +1) Corollary:(x − 1)|(xn − 1) So for an − 1tobeprime,weneeda =2. Moreover, if n = md, we can apply the lemma with x = ad.Then (ad − 1)|(an − 1) So we get the following Lemma If an − 1 is a prime, then a =2andn is prime. Definition:AMersenne prime is a prime of the form q =2p − 1,pprime. Question: are they infinitely many Mersenne primes? Best known: The 37th Mersenne prime q is associated to p = 3021377, and this was done in 1998. One expects that p = 6972593 will give the next Mersenne prime; this is close to being proved, but not all the details have been checked. Definition: A positive integer n is perfect iff it equals the sum of all its (positive) divisors <n. Definition: σ(n)= d|n d (divisor function) So u is perfect if n = σ(u) − n, i.e. if σ(u)=2n. Well known example: n =6=1+2+3 Properties of σ: 1. σ(1) = 1 1 2. n is a prime iff σ(n)=n +1 p σ pj p ··· pj pj+1−1 3. If is a prime, ( )=1+ + + = p−1 4. (Exercise) If (n1,n2)=1thenσ(n1)σ(n2)=σ(n1n2) “multiplicativity”. -
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. -
The Resultant of the Cyclotomic Polynomials Fm(Ax) and Fn{Bx)
MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 1-6 The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn{bx) By Tom M. Apóstol Abstract. The resultant p(Fm(ax), Fn(bx)) is calculated for arbitrary positive integers m and n, and arbitrary nonzero complex numbers a and b. An addendum gives an extended bibliography of work on cyclotomic polynomials published since 1919. 1. Introduction. Let Fn(x) denote the cyclotomic polynomial of degree sp{ri) given by Fn{x)= f['{x - e2"ikl"), k=\ where the ' indicates that k runs through integers relatively prime to the upper index n, and <p{n) is Euler's totient. The resultant p{Fm, Fn) of any two cyclotomic poly- nomials was first calculated by Emma Lehmer [9] in 1930 and later by Diederichsen [21] and the author [64]. It is known that piFm, Fn) = 1 if (m, ri) = 1, m > n>l. This implies that for any integer q the integers Fmiq) and Fn{q) are rela- tively prime if im, ri) = 1. Divisibility properties of cyclotomic polynomials play a role in certain areas of the theory of numbers, such as the distribution of quadratic residues, difference sets, per- fect numbers, Mersenne-type numbers, and primes in residue classes. There has also been considerable interest lately in relations between the integers FJq) and F Jp), where p and q are distinct primes. In particular, Marshall Hall informs me that the Feit-Thompson proof [47] could be shortened by nearly 50 pages if it were known that F Jq) and F Jp) are relatively prime, or even if the smaller of these integers does not divide the larger. -
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. -
08 Cyclotomic Polynomials
8. Cyclotomic polynomials 8.1 Multiple factors in polynomials 8.2 Cyclotomic polynomials 8.3 Examples 8.4 Finite subgroups of fields 8.5 Infinitude of primes p = 1 mod n 8.6 Worked examples 1. Multiple factors in polynomials There is a simple device to detect repeated occurrence of a factor in a polynomial with coefficients in a field. Let k be a field. For a polynomial n f(x) = cnx + ::: + c1x + c0 [1] with coefficients ci in k, define the (algebraic) derivative Df(x) of f(x) by n−1 n−2 2 Df(x) = ncnx + (n − 1)cn−1x + ::: + 3c3x + 2c2x + c1 Better said, D is by definition a k-linear map D : k[x] −! k[x] defined on the k-basis fxng by D(xn) = nxn−1 [1.0.1] Lemma: For f; g in k[x], D(fg) = Df · g + f · Dg [1] Just as in the calculus of polynomials and rational functions one is able to evaluate all limits algebraically, one can readily prove (without reference to any limit-taking processes) that the notion of derivative given by this formula has the usual properties. 115 116 Cyclotomic polynomials [1.0.2] Remark: Any k-linear map T of a k-algebra R to itself, with the property that T (rs) = T (r) · s + r · T (s) is a k-linear derivation on R. Proof: Granting the k-linearity of T , to prove the derivation property of D is suffices to consider basis elements xm, xn of k[x]. On one hand, D(xm · xn) = Dxm+n = (m + n)xm+n−1 On the other hand, Df · g + f · Dg = mxm−1 · xn + xm · nxn−1 = (m + n)xm+n−1 yielding the product rule for monomials. -
Mersenne and Fermat Numbers 2, 3, 5, 7, 13, 17, 19, 31
MERSENNE AND FERMAT NUMBERS RAPHAEL M. ROBINSON 1. Mersenne numbers. The Mersenne numbers are of the form 2n — 1. As a result of the computation described below, it can now be stated that the first seventeen primes of this form correspond to the following values of ra: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281. The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127—1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald [l]; see also Kraitchik [4]. The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer [7] and discussed by Uhler [13]. It is clear that 2" —1 can be factored algebraically if ra is composite; hence 2n —1 cannot be prime unless w is prime. Fermat's theorem yields a factor of 2n —1 only when ra + 1 is prime, and hence does not determine any additional cases in which 2"-1 is known to be com- posite. On the other hand, it follows from Euler's criterion that if ra = 0, 3 (mod 4) and 2ra + l is prime, then 2ra + l is a factor of 2n— 1. -
Primes and Primality Testing
Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100. -
20. Cyclotomic III
20. Cyclotomic III 20.1 Prime-power cyclotomic polynomials over Q 20.2 Irreducibility of cyclotomic polynomials over Q 20.3 Factoring Φn(x) in Fp[x] with pjn 20.4 Worked examples The main goal is to prove that all cyclotomic polynomials Φn(x) are irreducible in Q[x], and to see what happens to Φn(x) over Fp when pjn. The irreducibility over Q allows us to conclude that the automorphism group of Q(ζn) over Q (with ζn a primitive nth root of unity) is × Aut(Q(ζn)=Q) ≈ (Z=n) by the map a (ζn −! ζn) − a The case of prime-power cyclotomic polynomials in Q[x] needs only Eisenstein's criterion, but the case of general n seems to admit no comparably simple argument. The proof given here uses ideas already in hand, but also an unexpected trick. We will give a different, less elementary, but possibly more natural argument later using p-adic numbers and Dirichlet's theorem on primes in an arithmetic progression. 1. Prime-power cyclotomic polynomials over Q The proof of the following is just a slight generalization of the prime-order case. [1.0.1] Proposition: For p prime and for 1 ≤ e 2 Z the prime-power pe-th cyclotomic polynomial Φpe (x) is irreducible in Q[x]. Proof: Not unexpectedly, we use Eisenstein's criterion to prove that Φpe (x) is irreducible in Z[x], and the invoke Gauss' lemma to be sure that it is irreducible in Q[x]. Specifically, let f(x) = Φpe (x + 1) 269 270 Cyclotomic III p If e = 1, we are in the familiar prime-order case. -
Appendix a Tables of Fermat Numbers and Their Prime Factors
Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard -
Arxiv:0709.4112V1 [Math.NT] 26 Sep 2007 H Uhri Pnrdb H Okwgnfoundation
Elle est `atoi cette chanson Toi l’professeur qui sans fa¸con, As ouvert ma petite th`ese Quand mon espoir manquait de braise1. To the memory of Manuel Bronstein CYCLOTOMY PRIMALITY PROOFS AND THEIR CERTIFICATES PREDA MIHAILESCU˘ Abstract. The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007. Contents 1. Introduction 2 1.1. Some notations 5 2. Galois extensions of rings and cyclotomy 5 2.1. Finding Roots in Cyclotomic Extensions 11 2.2. Finding Roots of Unity and the Lucas – Lehmer Test 12 arXiv:0709.4112v1 [math.NT] 26 Sep 2007 3. GaussandJacobisumsoverCyclotomicExtensionsofRings 14 4. Further Criteria for Existence of Cyclotomic Extensions 17 5. Certification 19 5.1. Computation of Jacobi Sums and their Certification 21 6. Algorithms 23 7. Deterministic primality test 25 8. Asymptotics and run times 29 References 30 1 “Chanson du professeur”, free after G. Brassens Date: Version 2.0 October 26, 2018. The author is sponored by the Volkswagen Foundation. 1 2 PREDA MIHAILESCU˘ 1. Introduction Let n be an integer about which one wishes a decision, whether it is prime or not. The decision may be taken by starting from the definition, thus performing trial division by integers √n or is using some related sieve method, when the decision on a larger set of≤ integers is expected. -
Primality Testing for Beginners
STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.