Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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”. -
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. -
On Perfect and Multiply Perfect Numbers
On perfect and multiply perfect numbers . by P. ERDÖS, (in Haifa . Israel) . Summary. - Denote by P(x) the number o f integers n ~_ x satisfying o(n) -- 0 (mod n.), and by P2 (x) the number of integers nix satisfying o(n)-2n . The author proves that P(x) < x'314:4- and P2 (x) < x(t-c)P for a certain c > 0 . Denote by a(n) the sum of the divisors of n, a(n) - E d. A number n din is said to be perfect if a(n) =2n, and it is said to be multiply perfect if o(n) - kn for some integer k . Perfect numbers have been studied since antiquity. l t is contained in the books of EUCLID that every number of the form 2P- ' ( 2P - 1) where both p and 2P - 1 are primes is perfect . EULER (1) proved that every even perfect number is of the above form . It is not known if there are infinitely many even perfect numbers since it is not known if there are infinitely many primes of the form 2P - 1. Recently the electronic computer of the Institute for Numerical Analysis the S .W.A .C . determined all primes of the form 20 - 1 for p < 2300. The largest prime found was 2J9 "' - 1, which is the largest prime known at present . It is not known if there are an odd perfect numbers . EULER (') proved that all odd perfect numbers are of the form (1) pam2, p - x - 1 (mod 4), and SYLVESTER (') showed that an odd perfect number must have at least five distinct prime factors . -
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 PERFECT and NEAR-PERFECT NUMBERS 1. Introduction a Perfect
ON PERFECT AND NEAR-PERFECT NUMBERS PAUL POLLACK AND VLADIMIR SHEVELEV Abstract. We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation 12 = 1 + 2 + 3 + 6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpi´nski. We discuss some rules for generating near-perfect numbers similar to Euclid's rule for constructing even perfect numbers, and we obtain an upper bound of x5=6+o(1) for the number of near-perfect numbers in [1; x], as x ! 1. 1. Introduction A perfect number is a positive integer equal to the sum of its proper positive divisors. Let σ(n) denote the sum of all of the positive divisors of n. Then n is a perfect number if and only if σ(n) − n = n, that is, σ(n) = 2n. The first four perfect numbers { 6, 28, 496, and 8128 { were known to Euclid, who also succeeded in establishing the following general rule: Theorem A (Euclid). If p is a prime number for which 2p − 1 is also prime, then n = 2p−1(2p − 1) is a perfect number. It is interesting that 2000 years passed before the next important result in the theory of perfect numbers. In 1747, Euler showed that every even perfect number arises from an application of Euclid's rule: Theorem B (Euler). All even perfect numbers have the form 2p−1(2p − 1), where p and 2p − 1 are primes. -
+P.+1 P? '" PT Hence
1907.] MULTIPLY PEKFECT NUMBERS. 383 A TABLE OF MULTIPLY PERFECT NUMBERS. BY PEOFESSOR R. D. CARMICHAEL. (Read before the American Mathematical Society, February 23, 1907.) A MULTIPLY perfect number is one which is an exact divisor of the sum of all its divisors, the quotient being the multiplicity.* The object of this paper is to exhibit a method for determining all such numbers up to 1,000,000,000 and to give a complete table of them. I include an additional table giving such other numbers as are known to me to be multiply perfect. Let the number N, of multiplicity m (m> 1), be of the form where pv p2, • • •, pn are different primes. Then by definition and by the formula for the sum of the factors of a number, we have +P? -1+---+p + ~L pi" + Pi»'1 + • • (l)m=^- 1 •+P.+1 p? '" PT Hence (2) Pi~l Pa"1 Pn-1 These formulas will be of frequent use throughout the paper. Since 2•3•5•7•11•13•17•19•23 = 223,092,870, multiply perfect numbers less than 1,000,000,000 contain not more than nine different prime factors ; such numbers, lacking the factor 2, contain not more than eight different primes ; and, lacking the factors 2 and 3, they contain not more than seven different primes. First consider the case in which N does not contain either 2 or 3 as a factor. By equation (2) we have (^\ «M ^ 6 . 7 . 11 . 1£. 11 . 19 .2ft « 676089 [Ö) m<ï*6 TTT if 16 ï¥ ïî — -BTÏTT6- Hence m=2 ; moreover seven primes are necessary to this value. -
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. -
Some New Results on Odd Perfect Numbers
Pacific Journal of Mathematics SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT,JOHN L. HUNSUCKER AND CARL POMERANCE Vol. 57, No. 2 February 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 57, No. 2, 1975 SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT, J. L. HUNSUCKER AND CARL POMERANCE If ra is a multiply perfect number (σ(m) = tm for some integer ί), we ask if there is a prime p with m = pan, (pa, n) = 1, σ(n) = pα, and σ(pa) = tn. We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers neces- sarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) ~ 2n. 1* Introduction* In this paper we answer a question on odd perfect numbers posed by Suryanarayana [17]. It is known that if m is an odd perfect number, then m = pak2 where p is a prime, p Jf k, and p = a z= 1 (mod 4). Suryanarayana asked if it necessarily followed that (1) σ(k2) = pa , σ(pa) = 2k2 . Here, σ is the sum of the divisors function. We answer this question in the negative by showing that no odd perfect number satisfies (1). We actually consider a more general question. If m is multiply perfect (σ(m) = tm for some integer t), we say m has property S if there is a prime p with m = pan, (pa, n) = 1, and the equations (2) σ(n) = pa , σ(pa) = tn hold. -
Square Roots and the Rabin Cryptosystem William J
The Roots of Higher Mathematics Computing Square Roots and the Rabin Cryptosystem William J. Martin, WPI Abstract: We explore thought experiments dealing with the computation of square roots in various number systems. As an application, we introduce an encryption scheme of Rabin. 1 Weird Number Systems A child learns about number systems in a very sensible way: First the counting numbers, then perhaps integers, then fraction, on to real numbers, and then the complex numbers. This seems very reasonable: each contains all the previous ones and enjoys more and more features, allowing more and more flexibility. Before we discuss other systems, let us review the notation for these familiar objects. The natural numbers, or positive integers, are denoted N = f1; 2; 3;:::g. They are closed under addition and multiplication: if we add or multiply two numbers from this set, the answer is guaranteed to be in this set. This is a great context for a discussion of unique factorization: an integer p > 1 is prime if its only divisors are 1 and itself. (Note that one is neither prime nor composite; it has an inverse, so it is called a \unit".) The integers are the first ring a student sees. The set Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g is closed under both addition and multiplication. It is a commutative ring since ab = ba for any two integers a and b. Every element has an additive inverse: stately precisely, for any integer a, there is a (unique) integer b such that a + b = 0. In algebra, we extend intuitive notation and denote this b by \−a". -
SUPERHARMONIC NUMBERS 1. Introduction If the Harmonic Mean Of
MATHEMATICS OF COMPUTATION Volume 78, Number 265, January 2009, Pages 421–429 S 0025-5718(08)02147-9 Article electronically published on September 5, 2008 SUPERHARMONIC NUMBERS GRAEME L. COHEN Abstract. Let τ(n) denote the number of positive divisors of a natural num- ber n>1andletσ(n)denotetheirsum.Thenn is superharmonic if σ(n) | nkτ(n) for some positive integer k. We deduce numerous properties of superharmonic numbers and show in particular that the set of all superhar- monic numbers is the first nontrivial example that has been given of an infinite set that contains all perfect numbers but for which it is difficult to determine whether there is an odd member. 1. Introduction If the harmonic mean of the positive divisors of a natural number n>1isan integer, then n is said to be harmonic.Equivalently,n is harmonic if σ(n) | nτ(n), where τ(n)andσ(n) denote the number of positive divisors of n and their sum, respectively. Harmonic numbers were introduced by Ore [8], and named (some 15 years later) by Pomerance [9]. Harmonic numbers are of interest in the investigation of perfect numbers (num- bers n for which σ(n)=2n), since all perfect numbers are easily shown to be harmonic. All known harmonic numbers are even. If it could be shown that there are no odd harmonic numbers, then this would solve perhaps the most longstand- ing problem in mathematics: whether or not there exists an odd perfect number. Recent articles in this area include those by Cohen and Deng [3] and Goto and Shibata [5]. -
A Study of .Perfect Numbers and Unitary Perfect
CORE Metadata, citation and similar papers at core.ac.uk Provided by SHAREOK repository A STUDY OF .PERFECT NUMBERS AND UNITARY PERFECT NUMBERS By EDWARD LEE DUBOWSKY /I Bachelor of Science Northwest Missouri State College Maryville, Missouri: 1951 Master of Science Kansas State University Manhattan, Kansas 1954 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of .the requirements fqr the Degree of DOCTOR OF EDUCATION May, 1972 ,r . I \_J.(,e, .u,,1,; /q7Q D 0 &'ISs ~::>-~ OKLAHOMA STATE UNIVERSITY LIBRARY AUG 10 1973 A STUDY OF PERFECT NUMBERS ·AND UNITARY PERFECT NUMBERS Thesis Approved: OQ LL . ACKNOWLEDGEMENTS I wish to express my sincere gratitude to .Dr. Gerald K. Goff, .who suggested. this topic, for his guidance and invaluable assistance in the preparation of this dissertation. Special thanks.go to the members of. my advisory committee: 0 Dr. John Jewett, Dr. Craig Wood, Dr. Robert Alciatore, and Dr. Vernon Troxel. I wish to thank. Dr. Jeanne Agnew for the excellent training in number theory. that -made possible this .study. I wish tc;, thank Cynthia Wise for her excellent _job in typing this dissertation •. Finally, I wish to express gratitude to my wife, Juanita, .and my children, Sondra and David, for their encouragement and sacrifice made during this study. TABLE OF CONTENTS Chapter Page I. HISTORY AND INTRODUCTION. 1 II. EVEN PERFECT NUMBERS 4 Basic Theorems • • • • • • • • . 8 Some Congruence Relations ••• , , 12 Geometric Numbers ••.••• , , , , • , • . 16 Harmonic ,Mean of the Divisors •. ~ ••• , ••• I: 19 Other Properties •••• 21 Binary Notation. • •••• , ••• , •• , 23 III, ODD PERFECT NUMBERS . " . 27 Basic Structure • , , •• , , , . -
The Schnirelmann Density of the Set of Deficient Numbers
THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Peter Gerralld Banda 2015 SIGNATURE PAGE THESIS: THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS AUTHOR: Peter Gerralld Banda DATE SUBMITTED: Summer 2015 Mathematics and Statistics Department Dr. Mitsuo Kobayashi Thesis Committee Chair Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics Dr. John Rock Mathematics & Statistics ii ACKNOWLEDGMENTS I would like to take a moment to express my sincerest appreciation of my fianc´ee’s never ceasing support without which I would not be here. Over the years our rela tionship has proven invaluable and I am sure that there will be many more fruitful years to come. I would like to thank my friends/co-workers/peers who shared the long nights, tears and triumphs that brought my mathematical understanding to what it is today. Without this, graduate school would have been lonely and I prob ably would have not pushed on. I would like to thank my teachers and mentors. Their commitment to teaching and passion for mathematics provided me with the incentive to work and succeed in this educational endeavour. Last but not least, I would like to thank my advisor and mentor, Dr. Mitsuo Kobayashi. Thanks to his patience and guidance, I made it this far with my sanity mostly intact. With his expertise in mathematics and programming, he was able to correct the direction of my efforts no matter how far they strayed.