Multiplicative Series, Modular Forms, and Mandelbrot Polynomials
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Power Values of Divisor Sums Author(S): Frits Beukers, Florian Luca, Frans Oort Reviewed Work(S): Source: the American Mathematical Monthly, Vol
Power Values of Divisor Sums Author(s): Frits Beukers, Florian Luca, Frans Oort Reviewed work(s): Source: The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.119.05.373 . Accessed: 15/02/2013 04:05 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded on Fri, 15 Feb 2013 04:05:44 AM All use subject to JSTOR Terms and Conditions Power Values of Divisor Sums Frits Beukers, Florian Luca, and Frans Oort Abstract. We consider positive integers whose sum of divisors is a perfect power. This prob- lem had already caught the interest of mathematicians from the 17th century like Fermat, Wallis, and Frenicle. In this article we study this problem and some variations. We also give an example of a cube, larger than one, whose sum of divisors is again a cube. 1. INTRODUCTION. Recently, one of the current authors gave a mathematics course for an audience with a general background and age over 50. -
A Simple Proof of the Mean Fourth Power Estimate
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze K. RAMACHANDRA A simple proof of the mean fourth power estimate for 1 1 z(2 + it) and L(2 + it;c) Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 1, no 1-2 (1974), p. 81-97 <http://www.numdam.org/item?id=ASNSP_1974_4_1_1-2_81_0> © Scuola Normale Superiore, Pisa, 1974, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A Simple Proof of the Mean Fourth Power Estimate for 03B6(1/2 + it) and L(1/2 + it, ~). K. RAMACHANDRA (*) To the memory of Professor ALBERT EDWARD INGHAM 1. - Introduction. The main object of this paper is to prove the following four well-known theorems by a simple method. THEOREM 1. and then THEOREM 2. If T ~ 3, R>2 and Ti and also 1, then THEOREM 3. Let y be a character mod q (q fixed), T ~ 3, - tx,2 ... T, (.Rx > 2), and tx.i+,- 1. If with each y we associate such points then, where * denotes the sum over primitive character mod q. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
Statistics of Random Permutations and the Cryptanalysis of Periodic Block Ciphers
c de Gruyter 2008 J. Math. Crypt. 2 (2008), 1–20 DOI 10.1515 / JMC.2008.xxx Statistics of Random Permutations and the Cryptanalysis Of Periodic Block Ciphers Nicolas T. Courtois, Gregory V. Bard, and Shaun V. Ault Communicated by xxx Abstract. A block cipher is intended to be computationally indistinguishable from a random permu- tation of appropriate domain and range. But what are the properties of a random permutation? By the aid of exponential and ordinary generating functions, we derive a series of collolaries of interest to the cryptographic community. These follow from the Strong Cycle Structure Theorem of permu- tations, and are useful in rendering rigorous two attacks on Keeloq, a block cipher in wide-spread use. These attacks formerly had heuristic approximations of their probability of success. Moreover, we delineate an attack against the (roughly) millionth-fold iteration of a random per- mutation. In particular, we create a distinguishing attack, whereby the iteration of a cipher a number of times equal to a particularly chosen highly-composite number is breakable, but merely one fewer round is considerably more secure. We then extend this to a key-recovery attack in a “Triple-DES” style construction, but using AES-256 and iterating the middle cipher (roughly) a million-fold. It is hoped that these results will showcase the utility of exponential and ordinary generating functions and will encourage their use in cryptanalytic research. Keywords. Generating Functions, EGF, OGF, Random Permutations, Cycle Structure, Cryptanaly- sis, Iterations of Permutations, Analytic Combinatorics, Keeloq. AMS classification. 05A15, 94A60, 20B35, 11T71. 1 Introduction The technique of using a function of a variable to count objects of various sizes, using the properties of multiplication and addition of series as an aid, is accredited to Pierre- Simon Laplace [12]. -
Enciclopedia Matematica a Claselor De Numere Întregi
THE MATH ENCYCLOPEDIA OF SMARANDACHE TYPE NOTIONS vol. I. NUMBER THEORY Marius Coman INTRODUCTION About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name), it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. Because this is too vast to be covered in one book, we divide encyclopedia in more volumes. In this first volume of encyclopedia we try to synthesize his work in the field of number theory, one of the great Smarandache’s passions, a surfer on the ocean of numbers, to paraphrase the title of the book Surfing on the ocean of numbers – a few Smarandache notions and similar topics, by Henry Ibstedt. We quote from the introduction to the Smarandache’work “On new functions in number theory”, Moldova State University, Kishinev, 1999: “The performances in current mathematics, as the future discoveries, have, of course, their beginning in the oldest and the closest of philosophy branch of nathematics, the number theory. Mathematicians of all times have been, they still are, and they will be drawn to the beaty and variety of specific problems of this branch of mathematics. Queen of mathematics, which is the queen of sciences, as Gauss said, the number theory is shining with its light and attractions, fascinating and facilitating for us the knowledge of the laws that govern the macrocosm and the microcosm”. -
1.6 Exponents and the Order of Operations
1.6 EXPONENTS AND THE ORDER OF OPERATIONS Writing Whole Numbers and Variables in Exponent Form Student Learning Objectives Recall that in the multiplication problem 3 # 3 # 3 # 3 # 3 = 243 the number 3 is called a factor. We can write the repeated multiplication 3 # 3 # 3 # 3 # 3 using a shorter nota- After studying this section, you will tion, 35, because there are five factors of 3 in the repeated multiplication. We say be able to: 5 5 that 3 is written in exponent form. 3 is read “three to the fifth power.” Write whole numbers and variables in exponent form. Evaluate numerical and EXPONENT FORM algebraic expressions in The small number 5 is called an exponent. Whole number exponents, except exponent form. zero, tell us how many factors are in the repeated multiplication. The number Use symbols and key words 3 is called the base. The base is the number that is multiplied. for expressing exponents. Follow the order of operations. 3 # 3 # 3 # 3 # 3 = 35 The exponent is 5. 3 appears as a factor 5 times. The base is 3. We do not multiply the base 3 by the exponent 5. The 5 just tells us how many 3’s are in the repeated multiplication. If a whole number or variable does not have an exponent visible, the exponent is understood to be 1. 9 = 91 and x = x1 EXAMPLE 1 Write in exponent form. (a)2 # 2 # 2 # 2 # 2 # 2 (b)4 # 4 # 4 # x # x (c) 7 (d) y # y # y # 3 # 3 # 3 # 3 Solution # # # # # = 6 # # # x # x = 3 # x2 3x2 (a)2 2 2 2 2 2 2 (b) 4 4 4 4 or 4 (c)7 = 71 (d) y # y # y # 3 # 3 # 3 # 3 = y3 # 34, or 34y3 Note, it is standard to write the number before the variable in a term. -
MATHCOUNTS Bible Completed I. Squares and Square Roots: from 1
MATHCOUNTS Bible Completed I. Squares and square roots: From 12 to 302. Squares: 1 - 50 12 = 1 262 = 676 22 = 4 272 = 729 32 = 9 282 = 784 42 = 16 292 = 841 52 = 25 302 = 900 62 = 36 312 = 961 72 = 49 322 = 1024 82 = 64 332 = 1089 92 = 81 342 = 1156 102 = 100 352 = 1225 112 = 121 362 = 1296 122 = 144 372 = 1369 132 = 169 382 = 1444 142 = 196 392 = 1521 152 = 225 402 = 1600 162 = 256 412 = 1681 172 = 289 422 = 1764 182 = 324 432 = 1849 192 = 361 442 = 1936 202 = 400 452 = 2025 212 = 441 462 = 2116 222 = 484 472 = 2209 232 = 529 482 = 2304 242 = 576 492 = 2401 25 2 = 625 50 2 = 2500 II. Cubes and cubic roots: From 13 to 123. Cubes: 1 -15 13 = 1 63 = 216 113 = 1331 23 = 8 73 = 343 123 = 1728 33 = 27 83 = 512 133 = 2197 43 = 64 93 = 729 143 = 2744 53 = 125 103 = 1000 153 = 3375 III. Powers of 2: From 21 to 212. Powers of 2 and 3 21 = 2 31 = 3 22 = 4 32 = 9 23 = 8 33 = 27 24 = 16 34 = 81 25 = 32 35 = 243 26 = 64 36 = 729 27 = 128 37 = 2187 28 = 256 38 = 6561 29 = 512 39 = 19683 210 = 1024 310 = 59049 211 = 2048 311 = 177147 212 = 4096 312 = 531441 IV. Prime numbers from 2 to 109: It also helps to know the primes in the 100's, like 113, 127, 131, ... It's important to know not just the primes, but why 51, 87, 91, and others are not primes. -
Elementary Number Theory and Its Applications
Elementary Number Theory andlts Applications KennethH. Rosen AT&T Informotion SystemsLaboratories (formerly part of Bell Laborotories) A YY ADDISON-WESLEY PUBLISHING COMPANY Read ing, Massachusetts Menlo Park, California London Amsterdam Don Mills, Ontario Sydney Cover: The iteration of the transformation n/2 if n T(n) : \ is even l Qn + l)/2 if n is odd is depicted.The Collatz conjectureasserts that with any starting point, the iteration of ?"eventuallyreaches the integer one. (SeeProblem 33 of Section l.2of the text.) Library of Congress Cataloging in Publication Data Rosen, Kenneth H. Elementary number theory and its applications. Bibliography: p. Includes index. l. Numbers, Theory of. I. Title. QA24l.R67 1984 512',.72 83-l1804 rsBN 0-201-06561-4 Reprinted with corrections, June | 986 Copyright O 1984 by Bell Telephone Laboratories and Kenneth H. Rosen. 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 prior written permission of the publisher. printed in the United States of America. Published simultaneously in Canada. DEFGHIJ_MA_8987 Preface Number theory has long been a favorite subject for studentsand teachersof mathematics. It is a classical subject and has a reputation for being the "purest" part of mathematics, yet recent developments in cryptology and computer science are based on elementary number theory. This book is the first text to integrate these important applications of elementary number theory with the traditional topics covered in an introductory number theory course. This book is suitable as a text in an undergraduatenumber theory courseat any level. -
Proofs of Certain Arithmetic Theorems ∗
Proofs of certain arithmetic theorems ∗ Leonhard Euler Arithmetic theorems, of which Fermat and others unveiled many, are wor- thy of greater attention because of this, but instead their truth is hidden and difficult to prove. Indeed, Fermat has left behind a sufficiently great abundance of such theorems, but never published the proofs, even though he may have steadfastly claimed for himself to have established their validity with great cer- tainty. Therefore, it is very painful to write of the tragedy that, thus far, all of these proofs still remain unknown. The idea behind propositions in common knowledge is similar: in them it is claimed that neither the sum nor the dif- ference of fourth powers can be a square; though no one doubts the truth of this, a rigorous proof is still not available anywhere, as far as I know, except in a certain pamphlet once edited by Fr´enicle,the title of which is Trait´edes triangles rectangles en nombres. Here, the author proves among other things that in no right triangle whose sides are expressed as rational numbers can the area be a square, from which the truth of the previously mentioned proposi- tions about the sums and differences of two fourth powers is deduced. But that proof is very much enmeshed in the properties of triangles, so that unless the greatest attention is employed, it can be understood clearly only with great difficulty. On account of this, I believe these works to be of value, if I will have drawn the proofs of these propositions away from right triangles and put them forward clearly and analytically. -
It Is Wise, Generalize! by K
It is Wise, Generalize! by K. R. S. Sastry, Karnataka, India Often it is the case that the solution of one problem enables us to propose several new problems. This may take the form of a converse problem, an extension of the original problem in one of several directions or an analogy. By all means do consider them. It provides us with more problems to solve on the one hand. On the other, we generally employ different solution strategies to solve the newly proposed problems so we gain greater perspectives and valuable insights. Let us look at a solution of the following problem in the light of preceding suggestions. (*) Find a sequence of distinct natural numbers with the property that the difference of squares of any two consecutive terms is a perfect cube. Editor's Note: Try the problem yourself before reading on! Solution. If we assume s , s , . as the sequence of natural numbers to be determined, then we must have s - s = t , r = 1, 2, 3, ..... (1) where t's are also natural numbers. When r = 1 then s - s = t . A little experimentation shows that 3 - 1 = 2 . So we take s = 1, s = 3, t = 2. When r = 2 then s - s = t holds. With s = 3, we find that s = 6. Likewise we find that s = 10. Let us write down the initial terms of the sequence: 1, 3, 6, 10, s , s , . It is now easy to guess that s = 15, s = 21, and that s = r(r + 1)/2. By using the induction principle we can convince ourselves that s = r(r + 1)/2 r = 1, 2, 3, . -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number) -
Finding Strong Pseudoprimes to Several Bases
MATHEMATICS OF COMPUTATION Volume 70, Number 234, Pages 863{872 S 0025-5718(00)01215-1 Article electronically published on February 17, 2000 FINDING STRONG PSEUDOPRIMES TO SEVERAL BASES ZHENXIANG ZHANG Dedicated to the memory of P. Erd}os (1913{1996) Abstract. Define m to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of m, we will have, for integers n< m, a deterministic primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pomerance et al. and Jaeschke, m are known for 1 ≤ m ≤ 8. Upper bounds for 9; 10 and 11 were given by Jaeschke. In this paper we tabulate all strong pseudoprimes (spsp's) n<1024 to the first ten prime bases 2; 3; ··· ; 29; which have the form n = pq with p; q odd primes and q −1=k(p−1);k=2; 3; 4: There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp's to both bases 31 and 37. As a result the upper bounds for 10 and 11 are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for 12 is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases.