Appendix a Tables of Fermat Numbers and Their Prime Factors
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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”. -
81684-1B SP.Pdf
INTERMEDIATE ALGEBRA The Why and the How First Edition Mathew Baxter Florida Gulf Coast University Bassim Hamadeh, CEO and Publisher Jess Estrella, Senior Graphic Designer Jennifer McCarthy, Acquisitions Editor Gem Rabanera, Project Editor Abbey Hastings, Associate Production Editor Chris Snipes, Interior Designer Natalie Piccotti, Senior Marketing Manager Kassie Graves, Director of Acquisitions Jamie Giganti, Senior Managing Editor Copyright © 2018 by Cognella, Inc. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of Cognella, Inc. For inquiries regarding permissions, translations, foreign rights, audio rights, and any other forms of reproduction, please contact the Cognella Licensing Department at [email protected]. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Cover image copyright © Depositphotos/agsandrew. Printed in the United States of America ISBN: 978-1-5165-0303-2 (pbk) / 978-1-5165-0304-9 (br) iii | CONTENTS 1 The Real Number System 1 RealNumbersandTheirProperties 2:1.1 RealNumbers 2 PropertiesofRealNumbers 6 Section1.1Exercises 9 TheFourBasicOperations 11:1.2 AdditionandSubtraction 11 MultiplicationandDivision -
On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer). -
Prime Divisors in the Rationality Condition for Odd Perfect Numbers
Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied. -
The Unique Natural Number Set and Distributed Prime Numbers
mp Co uta & tio d n e a li Jabur and Kushnaw, J Appl Computat Math 2017, 6:4 l p M p a Journal of A t h DOI: 10.4172/2168-9679.1000368 f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access The Unique Natural Number Set and Distributed Prime Numbers Alabed TH1* and Bashir MB2 1Computer Science, Shendi University, Shendi, Sudan 2Shendi University, Shendi, Sudan Abstract The Natural number set has a new number set behind three main subset: odd, even and prime which are unique number set. It comes from a mathematical relationship between the two main types even and odd number set. It consists of prime and T-semi-prime numbers set. However, finding ways to understand how prime number was distributed was ambiguity and it is impossible, this new natural set show how the prime number was distributed and highlight focusing on T-semi-prime number as a new sub natural set. Keywords: Prime number; T-semi-prime number; Unique number purely mathematical sciences and it has been used prime number set within theoretical physics to try and produce a theory that ties together prime numbers with quantum chaos theory [3]. However, researchers Introduction were made a big effort to find algorithm which can describe how prime The Unique natural number set is sub natural number set which number are distributed, still distribution of prime numbers throughout consist of prime and T-semi-prime numbers. T-semi-prime is natural a list of integers leads to the emergence of many unanswered and number divided by one or more prime numbers. -
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. -
A Member of the Order of Minims, Was a Philosopher, Mathematician, and Scientist Who Wrote on Music, Mechanics, and Optics
ON THE FORM OF MERSENNE NUMBERS PREDRAG TERZICH Abstract. We present a theorem concerning the form of Mersenne p numbers , Mp = 2 − 1 , where p > 3 . We also discuss a closed form expression which links prime numbers and natural logarithms . 1. Introduction. The friar Marin Mersenne (1588 − 1648), a member of the order of Minims, was a philosopher, mathematician, and scientist who wrote on music, mechanics, and optics . A Mersenne number is a number of p the form Mp = 2 − 1 where p is an integer , however many authors prefer to define a Mersenne number as a number of the above form but with p restricted to prime values . In this paper we shall use the second definition . Because the tests for Mersenne primes are relatively simple, the record largest known prime number has almost always been a Mersenne prime . 2. A theorem and closed form expression Theorem 2.1. Every Mersenne number Mp , (p > 3) is a number of the form 6 · k · p + 1 for some odd possitive integer k . Proof. In order to prove this we shall use following theorems : Theorem 2.2. The simultaneous congruences n ≡ n1 (mod m1) , n ≡ n2 (mod m2), .... ,n ≡ nk (mod mk) are only solvable when ni = nj (mod gcd(mi; mj)) , for all i and j . i 6= j . The solution is unique modulo lcm(m1; m2; ··· ; mk) . Proof. For proof see [1] Theorem 2.3. If p is a prime number, then for any integer a : ap ≡ a (mod p) . Date: January 4 , 2012. 1 2 PREDRAG TERZICH Proof. -
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. -
The Pseudoprimes to 25 • 109
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a. -
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. -
Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11
q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n.