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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 -
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. -
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. -
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. -
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. -
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 . -
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. -
Sequences of Primes Obtained by the Method of Concatenation
SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd