1 the Concept of Numbers
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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. -
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 -
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. -
2017 Grand Lodge of Minnesota Annual Communication Proceedings
2017 PROCEEDINGS The Grand Lodge A.F. and A.M. Minnesota Robert L. Darling, Grand Master Link to interactive index page 2017 ANNUAL PROCEEDINGS GRAND LODGE A. F. & A. M. of MINNESOTA 11501 Masonic Home Drive Bloomington, MN 55437-3699 952-948-6700 800-245-6050 952-948-6710 Fax E-Mail:[email protected] www.mn-masons.org 2017 ANNUAL PROCEEDINGS 3 ROBERT L. DARLING GRAND MASTER 4 GRAND LODGE OF MINNESOTA BIOGRAPHY GRAND MASTER ROBERT L. DARLING Robert L. Darling, “Bob”, was born on February 17, 1956 in Mattoon, Illinois. His parents were Russell D. and Theresa D. Darling. They lived in Greenup, Illinois. The family moved from Greenup to Decatur, Illinois and then to Maroa, Illinois where he attended the Maroa Elementary and Maroa-Forsyth High School. After graduating from the high school in mid-year, Bob enrolled and attended Illinois State University located in Normal, Illinois. In December 1976, he graduated with a B.S. Degree in Industrial Technology. Bob has worked for numerous companies including Caterpillar Inc. in Decatur, Illinois; Baldwin Associates, Clinton, Illinois; Schrock Cabinets/An Electrolux Company, Arthur, Illinois, Electrolux Home Products, St. Cloud, Minnesota. He is currently employed with the State of Minnesota, Department of Labor and Industry, OSHA Enforcement as a Safety Investigator Principal, and has worked there since 2003. Bob has been a Master Mason for 29 years. He was initiated on November 23, 1987; passed to a Fellowcraft on December 12, 1987; and was raised to the Sublime Degree of a Master Mason on January 9, 1988 by Maroa Lodge No. -
On Perfect Numbers and Their Relations 1 Introduction
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 27, 1337 - 1346 On Perfect Numbers and their Relations Recep G¨ur 2000 Evler Mahallesi Boykent Sitesi E Blok 26/7 Nevsehir, Turkey [email protected] Nihal Bircan Technische Universitaet Berlin Fakultaet II Institut f¨ur Mathematik MA 8 − 1 Strasse des 17. Juni 136 D-10623 Berlin, Germany [email protected] Abstract In this paper, we get some new formulas for generalized perfect numbers and their relationship between arithmetical functions φ, σ concerning Ore’s harmonic numbers and by using these formulas we present some examples. Mathematics Subject Classification: 11A25, 11A41, 11Y70 Keywords: perfect number, 2-hyperperfect number, Euler’s totient function, Ore harmonic number 1 Introduction In this section, we aimed to provide general information on perfect numbers shortly. Here and throughout the paper we assume m, n, k, d, b are positive integers and p is prime. N is called perfect number, if it is equal to the sum of its proper divisors. The first few perfect numbers are 6, 28, 496, 8128,... since, 6 = 1+2+3 28 = 1+2+4+7+14 496 = 1+2+4+8+16+31+62+124+248 Euclid discovered the first four perfect numbers which are generated by the formula 2n−1(2n−1), called Euclid number. In his book ’Elements’ he presented the proof of the formula which gives an even perfect number whenever 2n −1is 1338 Recep G¨ur and Nihal Bircan prime. In order for 2n −1 to be a prime n must itself be a prime. -
Euclid's Number Theory
Euclid of Alexandria, II: Number Theory Euclid of Alexandria, II: Number Theory Waseda University, SILS, History of Mathematics Euclid of Alexandria, II: Number Theory Outline Introduction Euclid’s number theory The overall structure Definitions for number theory Theory of prime numbers Properties of primes Infinitude of primes Euclid of Alexandria, II: Number Theory Introduction Concepts of number § The natural numbers is the set N “ t1; 2; 3;::: u. § The whole numbers is the set W “ t0; 1; 2; 3;::: u. § The integers is the set of positive and negative whole numbers Z “ t0; 1; ´1; 2; ´2;::: u. § The rational numbers is the set Q, of numbers of the form p{q, where p; q P Z,1 but q ‰ 0. § The real numbers, R, is the set of all the values mapped to the points of the number line. (The definition is tricky.) § An irrational number is a number that belongs to the reals, but is not rational. 1The symbol P means “in the set of,” or “is an element of.” Euclid of Alexandria, II: Number Theory Introduction The Greek concept of number § Greek number theory was exclusively interested in natural numbers. § In fact, the Greek also did not regard “1” as a number, but rather considered it the unit by which other numbers are numbered (or measured). § We can define Greek natural numbers as G “ t2; 3; 4;::: u. (But we can do most Greek number theory with N, so we will generally use this set, for simplicity.) Euclid of Alexandria, II: Number Theory Introduction Number theory before Euclid § The semi-legendary Pythagorus himself and other Pythagoreans are attributed with a fascination with numbers and with the development of a certain “pebble arithmetic” which studied the mathematical properties of numbers that correspond to certain geometry shapes (figurate numbers). -
Euclid's Proof of the Infinitude of Primes
Euclid’s Proof of the Infinitude of Primes: Distorted, Clarified, Made Obsolete, and Confirmed in Modern Mathematics REINHARD SIEGMUND-SCHULTZE n this article1 I reflect on the recurrent theme of Michael Hardy and Catherine Woodgold (henceforth modernizing historical mathematical proofs, vocabulary, H&W) entitled ‘‘Prime Simplicity’’ appeared in this journal II and symbolism, and the extent to which this modern- in 2009. H&W discuss Euclid’s proof mainly from the ization serves to clarify, is able to preserve, or is bound to point of view of its distortion by modern authors who distort the original meaning. My example is also a quite often claim that it had the form of a reductio ad recurrent one: Euclid’s proof of the infinitude of primes absurdum.3 in book IX, theorem 20 of his Elements. Elementary My article, on the other hand, aims at a systematic pre- number theory is a very appropriate field for discussing sentation and logical analysis of Euclid’s proof (as it is such general historiographic questions on a nontechnical preserved in the critical editions by Euclid scholars) and, level.2 Indeed, a widely quoted and stimulating article by above all, at a more detailed discussion of its interpretation 1I dedicate this article to Walter Purkert (Bonn), the coordinator and very active force behind the excellent Felix Hausdorff edition soon to be completed, on the occasion of his 70th birthday in 2014. 2Mesˇ trovic´ (2012) contains many details on modern proofs of the infinitude of primes but is less interested in Euclid’s original theorem and its modern variants. -
Mathematical Constants and Sequences
Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) .. -
Survey of Math: Chapter 21: Consumer Finance Savings Page 1
Survey of Math: Chapter 21: Consumer Finance Savings Page 1 Geometric Series Consider the following series: 1 + x + x2 + x3 + ··· + xn−1. We need to figure out a way to write this without the ···. Here’s how: s = 1 + x + x2 + x3 + ··· + xn−1 subtract xs = x + x2 + x3 + ··· + xn−1 + xn s − sx = 1 − 0 − 0 − 0 − · · · − 0 − xn s − sx = 1 − xn Now solve for s: 1 − xn xn − 1 s(1 − x) = 1 − xn ⇒ s = = . 1 − x x − 1 xn − 1 Therefore, a geometric series has the following sum: 1 + x + x2 + x3 + ··· + xn−1 = . x − 1 Exponential and Natural Logarithms As we have seen, some of our equations involve exponents. To effectively deal with exponents, we need to be able to work with exponentials and logarithms. The exponential and natural logarithm functions are inverse functions, and related by the following rules 1 m e = 1 + for m large m e = 2.71828 ... ln(eA) = A where A is a constant eln(A) = A If the base of the exponent is not e, we can use the following rule: ln(bA) = A ln(b) where A and b are constants ln(2A) = A ln(2) ln((1 + r)A) = A ln(1 + r) Note: • The natural logarithm acts on a number, so we read ln(2) as “The natural logarithm of 2”. • We would never write ln(2) = 2 ln, since this loses the fact that the natural logarithm must act on something. It is a functional evaluation, not a multiplication. • This is similar to trigonometric functions, like sin(π). -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
Introduction to the Theory of Numbers (Chicago University Press, 1929 : Introduction)
AN INTRODUCTION TO THE THEORY OF NUMBERS AN INTRODUCTION TO THE THEORYOFNUMBERS BY G. H. HARDY AND E. M. WRIGHT Principal and Vice-Chancellor of the University of Aberdeen FOURTH EDITION OX.FORD AT THE CLARENDON PRESS Ox@d University Press, Ely House, London W. 1 OLASOOWNEW YORK TORONTOMELBOURNE WELLINGTON CAPETOWN IBADAN NAIROBI DAR ES SALAAMI.USAKA ADDISABABA DELEI BOMBAYC.4I.CUTTA MADRAS KARACHI LAHOREDACCA KUALALUMPUR SINOAPORE HONO RONO TOKYO ISBN 0 19 853310 7 Fi& edition 1938 Second edition xg45 Third edition 1954 Fourth edition 1960 rg6z (with corrections) 1965 (with corrections) 1968 (with cowectiona) =97=> 1975 Printed in Great B&ain at the University Press, Oxford by Vivian Ridler Printer to the University PREFACE TO THE FOURTH EDITION APART from the provision of an index of names, the main changes in this edition are in the Notes at the end of each chapter. These have been revised to include references to results published since the third edition went to press and to correct omissions. Therc are simpler proofs of Theorems 234, 352, and 357 and a new Theorem 272. The Postscript to the third edition now takes its proper place as part of Chapter XX. 1 am indebted to several correspondents who suggested improvements and corrections. 1 have to thank Dr. Ponting for again reading the proofs and Mrs. V. N. R. Milne for compiling the index of names. E. M. W. ABERDEEN July 1959 PREFACE TO THE FIRST EDITION THIS book has developed gradually from lectures delivered in a number of universities during the last ten years, and, like many books which have grown out of lectures, it has no very definite plan. -
Calculating Logs Manually
calculating logs manually File Name: calculating logs manually.pdf Size: 2113 KB Type: PDF, ePub, eBook Category: Book Uploaded: 3 May 2019, 12:21 PM Rating: 4.6/5 from 561 votes. Status: AVAILABLE Last checked: 10 Minutes ago! In order to read or download calculating logs manually ebook, you need to create a FREE account. Download Now! eBook includes PDF, ePub and Kindle version ✔ Register a free 1 month Trial Account. ✔ Download as many books as you like (Personal use) ✔ Cancel the membership at any time if not satisfied. ✔ Join Over 80000 Happy Readers Book Descriptions: We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer, you have convenient answers with calculating logs manually . To get started finding calculating logs manually , you are right to find our website which has a comprehensive collection of manuals listed. Our library is the biggest of these that have literally hundreds of thousands of different products represented. Home | Contact | DMCA Book Descriptions: calculating logs manually And let’s start with the logarithm of 2. As a kid I always wanted to know how to calculate log 2 and nobody was able to tell me. Can we guess the logarithm of 19,683.Let’s follow the chain. We are looking for 1225, so to account for the difference let’s round 3.0876 up to 3.088. How to calculate log 11. Here is a way to do this. We can take the geometric mean of 10 and 12.