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Springer Series in Information Sciences 7
Springer-Verlag Berlin Heidelberg GmbH
Springer Series in Information SciencesEditors: Thomas S. Huang Teuvo Kohonen Manfred R. SchroederManaging Editor: H. K.V. Lotsch
30 Self-Organizing MapsBy T. Kohonen 2nd Edition
31 Music and Schema TheoryCognitive Foundationsof Systematic MusicologyBy M. Leman
32 The Maximum Entropy MethodByN. Wu
33 Steps Towards 3D Active VisionBy T. Vieville
34 Calibration and Orientation of Cameras in Computer VisionEditors: A. Griin and T. S. Huang
35 Speech Processing: Fundamentals and ApplicationsBy B. S. Atal and M. R. Schroeder
Volumes 1-29 are listed at the end of the book.
M. R. Schroeder
Number Theoryin Scienceand CommunicationWith Applications in Cryptography,Physics, Digital Information, Computing,and Self-Similarity
Third EditionWith 99 Figures
Springer
Professor Dr. Manfred R. Schroeder
Direktor, Drittes Ph ysikalisches Institut, Un iversität Göttingen, Bürge rst raß e 42- 44,D-37073 Göttingen , Germany andPast Director, Ac ou st ic s Speech and Mechanics Re sear ch , Bell Lab orat ories ,Murray Hili , NJ 07974, USA
Se ries Editors:
Professor Thomas S. HuangDep artment of Electrical Engineering and Coordinated Science Lab orat or y,Universit y of IIIinois, Urbana, IL 6 180 1, USA
Professor Teuvo KohonenHel sinki University of Technology , Neural Networks Research Centre, Rak entajanaukio 2 C,FIN-02 I 50 Espoo, Finland
Professor Dr. Manfred R. SchroederDrittes Physikalisches Institut, Universität Göttingen, Bürgerstrasse 42 -44 ,D-37073 Göttingen, Germany
Managing Editor:
Dr.-Ing. Helmut K. V. LotschSpringer-Verlag, Tiergarten strasse 17,D-69121 Heidelberg, German y
ISSN 0720-678XISBN 978-3-540-62006-8 ISBN 978-3-662-03430-9 (eBook)DOI 10.1007/978-3-662-03430-9
Library of Congress Cataloging-in-Publi cation Data.Schroeder, M. R. (Manfred Robert), 1926- Number theory in science and communication : with applications incryptography, physics, digital information, cornputing, and self-similarity / M. R. Schroeder. - 3rd ed. p. cm. (Springer series in information sciences, ISSN 0720-678X : 7) Includes bibliographieal references (p. - ) andindex. I. Numbertheory. I. Title. 11. Series. QA24l.S3 18 1997 512' .7- dc21 96-53994
This work is subject to copyright. All rights are reserved , whether the whole or part of the material is concerned,specifically the rights of translation, reprintin g, reuse of illustrations, recitation, broadcasting, reproducti on onmicrofilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof ispermitred only under the provisions of the German Copyright Law of September 9, 1965, in its current version,and perrnission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution underthe German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1984,1986,1997Originally published by Springer-Verlag Berlin Heidelberg New York in 1997.
The use of general descriptive names, registered names, trademarks, etc, in this publication does not imply, evenin the absence of a specific statement, that such names are exempt from the relevant protective laws andregulations and therefore free for general use.
Typesetting: Adam Leinz, Karlsruhe, by using Springer TEX rnacro package "c1monoOI"Cover design: design & production GmbH, HeidelbergSPIN: 10558990 54/3144 - 5 4 3 2 I 0 - Printed on acid-free paper
Dedicated to the Memory of
Hermann Minkowski
who added a fourth dimension to our Worldand many more to Number Theory
Foreword
"Beauty is the first test: there is nopermanent place in the world forugly mathematics."
- G. H. Hardy
Number theory has been considered since time immemorial to be the veryparadigm of pure (some would say useless) mathematics. In fact, the Chinesecharacters for mathematics are Number Science. "Mathematics is the queenof sciences - and number theory is the queen of mathematics", accordingto Carl Friedrich Gauss, the lifelong Wunderkind, who himself enjoyed theepithet "Princeps Mathematicorum". What could be more beautiful than adeep, satisfying relation between whole numbers? (One is almost tempted tocall them wholesome numbers.) In fact, it is hard to come up with a moreappropriate designation than their learned name: the integers - meaning the"untouched ones". How high they rank, in the realms of pure thought andaesthetics, above their lesser brethren: the real and complex numbers - whosefirst names virtually exude unsavory involvement with the complex realitiesof everyday life!
Yet, as we shall see in this book, the theory of integers can provide totallyunexpected answers to real-world problems. In fact , discrete mathematics istaking on an ever more important role. If nothing else, the advent of thedigital computer and digital communication has seen to that . But even earlier, in physics the emergence of quantum mechanics and discrete elementaryparticles put a premium on the methods and, indeed , the spirit of discretemathematics.
And even in mathematics proper, Hermann Minkowski, in the prefaceto his introductory book on number theory, Diophantische Approximationen, published in 1907 (the year he gave special relativity its proper fourdimensional clothing in preparation for its journey into general covarianceand cosmology) expressed his conviction that the "deepest interrelationshipsin analysis are of an arithmetical nature" .
Yet much of our schooling concentrates on analysis and other branchesof continuum mathematics to the virtual exclusion of number theory, grouptheory, combinatorics and graph theory. As an illustration, at a recent symposium on information theory, the author met several young researchers formally trained in mathematics and working in the field of primality testing,who - in all their studies up to the Ph.D. - had not heard a single lecture onnumber theory!
Or, to give an earlier example, when Werner Heisenberg discovered "matrix" mechanics in 1925, he didn't know what a matrix was (Max Born hadto tell him) , and neither Heisenberg nor Born knew what to make of the
VIII Foreword
appearance of matrices in the context of the atom. (David Hilbert is reported to have told them to go look for a differential equation with the sameeigenvalues, if that would make them happier. They did not follow Hilbert'swell-meant advice and thereby may have missed discovering the Schrodingerwave equation.)
The present book seeks to fill this gap in basic literacy in number theory- not in any formal way, for numerous excellent texts are available - but in amanner that stresses intuition and interrelationships, as well as applicationsin physics, biology, computer science, digital communication, cryptographyand more playful things, such as puzzles, teasers and artistic designs.
Among the numerous applications of number theory on which we willfocus in the subsequent chapters are the following:
1) The division of the circle into equal parts (a classical Greek preoccupation) and the implications of this ancient art for modern fast computationand random number generation.
2) The Chinese remainder theorem (another classic, albeit far Eastern)and how it allows us to do coin tossing over the telephone (and many thingsbesides).
3) The design of concert hall ceilings to scatter sound into broad lateralpatterns for improved acoustic quality (and wide-scatter diffraction gratingsin general).
4) The precision measurement of delays of radar echoes from Venus andMercury to confirm the general relativistic slowing of electromagnetic wavesin gravitational fields (the "fourth" - and last to be confirmed - effect predicted by Einstein's theory of general relativity).
5) Error-correcting codes (giving us distortion-free pictures of Jupiter andSaturn and their satellites).
6) "P ublic-key" encryption and deciphering of secret messages. Thesemethods also have important implications for computer security.
7) The creation of artistic graphic designs based on prime numbers.8) How to win at certain parlor games by putting the Fibonacci number
systems to work.9) The relations between Fibonacci numbers and the regular pentagon,
the Golden ratio, continued fractions, efficient approximations, electrical networks, the "squared" square, and so on - almost ad infinitum.
I dedicated this book to Hermann Minkowski because he epitomizes, in mymind, the belief in the usefulness outside mathematics of groups and numbertheory. He died young and never saw these concepts come to full fruition ingeneral relativity, quantum mechanics and some of the topics touched uponhere. I am therefore glad that the town of Cottingen is moving to honor itsformer resident on the occasion of the 100th anniversary of his doctorate on30 July, 1885 (under F. Lindemann, in now transcendental Konigsberg) . Thelate Lilly Riidenberg, nee Minkowski (born in Zurich, while her father wasteaching a still unknown Albert there), communicated valuable informationin preparation for this late recognition.
Preface to the Third Edition
Number theory has not rested on its laurels since the appearance of thesecond edition in 1985. Fermat's last theorem has finally been proven - some350 years after its bold pronouncement. And Andrew Wiles will before longreceive the coveted Wolfskehl Prize of the Oottingen Academy of Sciences .Elliptic curves, which played such a large role in the proof, have given newimpetus to fast factoring (Sect . 19.8) . Even faster factoring is in the offingif ever quantum computers can be persuaded to cohere long enough for theresults to be read out (Sect . 19.8) .
Additive number theory has found new applications in exact models instatistical mechanics. This new edition has therefore been amplified by several topics from additive number theory: prime clusters and prime spacings (Sects.4.8-12), the Goldbach conjecture (Sect .4.13), and the sum ofthree primes (Sect. 4.14) . "Golomb rulers", which started out as a curiosity,have found important applications in radio astronomy and signal processing(Sect. 28.6). A new application of the two-squares theorem allows the creationof circularly polarized acoustic waves (Sect. 7.9) .
Much has also happened in the application of number theory to dynamicalsystems (Sects . 5.10.1 and 5.10.2) .
Galois ("maximum-length") sequences have been turned into multidimensional arrays with surprising applications in X-ray astronomy (Sect. 13.10).New gratings and antenna arrays based on the number-theoretic logarithmshow directional patterns with suppressed broadside radiation (Sect. 15.10).
Finally, certain aperiodic sequences of integers have given rise to intriguingrhythms and "baroque" melodies that make us recall the Italian composerDomenico Scarlatti (Sect. 29.4).
Commensurate with its practical importance, the chapters on cryptography and data security have been amplified (Chaps. 9 and 12).
Since the appearance of the second edition several conferences have beendevoted to the practical applications of number theory, such as the LesHouches Winter School "Number Theory in Physics" [J. M. Luck et al. Number Theory in Physics, Springer 1990] and the symposium of the AmericanMathematical Society on the "Unreasonable Effectiveness of Number Theory" . [So A. Burr (Ed.) : The Unreasonable Effectiveness of Number Theory,Am. Math. Soc. 1991].
Cottingen and Berkeley HeightsMarch 1997
Manfred R. Schroeder
Preface to the Second Edition
The first edition quickly shrank on Springer's shelves, giving me a welcomeopportunity to augment this volume by some recent forays of number theoryinto new territory. The most exciting among these is perhaps the discovery,in 1984, of a new state of matter, sharing important properties with bothperfect crystals and amorphous substances, without being either one of these.The atomic structure of this new state is intimately related to the Goldenratio and a certain self-similar (rab)bit sequence that can be derived fromit . In fact, certain generalizations of the Golden ratio, the "Silver ratios" numbers that can be expressed as periodic continued fractions with periodlength one - lead one to postulate quasicrystals with "forbidden" 8- and12-fold symmetries and additional quasicrystals with 5-fold symmetry whoselattice parameters are generated by the Lucas numbers (Sect . 30.1).
The enormously fruitful concept of self-similarity, which pervades naturefrom the distribution of atoms in matter to that of the galaxies in the universe,also occurs in number theory. And because self-similarity is such a prettysubject, in which Cantor and Julia sets join Weierstrass functions to createa new form of art (distinguished by fractal Hausdorff dimensions), a briefcelebration of this strangely attractive union seems all but self set.
It is perhaps symptomatic that with set theory still another abstractbranch of mathematics has entered the real world. Who would have thoughtthat such utterly mathematical const ruct ions as Cantor sets, invented solelyto reassure the sceptics that sets could have both measure zero and still beuncountable, would make a difference in any practical arena, let alone becomea pivotal concept? Yet this is precisely what happened for many natural phenomena from gelation, polymerization and coagulation in colloidal physics tononlinear systems in many branches of science. Percolation, dendritic growth,electrical discharges (lightnings and Lichtenberg figures) and the compositionof glasses are best described by set-theoretic fractal dimensions.
Or take the weird functions Weierstrass invented a hundred years agopurely to prove that a function could be both everywhere continuous and yetnowhere differentiable. The fact that such an analytic pathology describessomething in the real world - nay, is elemental to understanding the basinsof attraction of strange attractors, for one - gives one pause. These excitingnew themes are sounded in Chap. 30.
XII Preface to the Second Edition
Physicists working in deterministic chaos have been touting the Goldenratio g as the "most irrational" of irrational numbers; and now they gladdenus with yet another kind of new number: the noble numbers, of which g (howaptly named!) is considered the noblest. Nonlinear dynamical systems governed by these "new" numbers (whose continued-fraction expansions end ininfinitely many l's) show the greatest resistance to the onslaught of chaoticmotion (such as turbulence). The rings of Saturn and, indeed, the very stability of the solar system are affected by these numbers. This noble feat, too,merits honorable mention (in Sects . 5.3 and 30.1).
Another topic, newly treated and of considerable contemporary import,is error-free computation, based on Farey fractions and p-adic "Hensel codes"(Sect. 5.12) .
Other recent advances recorded here are applications of the Zech logarithmto the design of optimum ambiguity functions for radar, new phase-arrayswith unique radiation patterns, and spread-spectrum communication systems(Sects. 25.7-9).
A forthcoming Italian translation occasioned the inclusion of a bankingpuzzle (Sect . 5.11) and other new material on Fibonacci numbers (in Sect. 1.1and Chap. 30).
Murray Hill and GottingenApril 1985
Manfred R. Schroeder
Acknowledgements
I thank Martin Kneser, Don Zagier, Henry Pollak, Jeff Lagarias and HansWerner Strube for their careful reviews of the manuscript .
Wilhelm Magnus, Armin Kohlrausch, Paul C. Spandikow, Herbert Taylorand Martin Gardner spotted several surviving slips.
Joseph L. Hall and AT&T Bell Laboratories computed the spectra of theself-similar sequences shown in Chap. 30. Peter Meyer, University of G6ttingen, prepared the auditory paradox, described in Sect. 30.2, based on a"fract al" signal waveform.
Norma Culviner's red pencil and yellow paper slips have purged the syntax of original sin. Angelika Weinrich, Irena Sch6nke and Elvire Hung, withlinguistic roots far from English soil, and Dorothy Caivano have typed pagesand fed computers to near system overflow. Gisela Kirschmann-Schroder andLiane Liebe prepared the illustrations with professional care and love. AnnySchroeder piloted the ms. through the sea of corrections inundating her deskfrom two continents.
Dr. Helmut Lotsch of Springer-Verlag deserves recognition for his leadership in scientific publication. Working with Reinhold Michels and ClausDieter Bachem from the production department in Heidelberg proved a puredelight.
William O. Baker and Arno Penzias fostered and maintained the researchclimate at Bell Laboratories in which an endeavor such as the writing of thisbook could flourish . Max Mathews provided support when it was needed.John R. Pierce inspired me to write by his writings.
Contents
Part I. A Few Fundamentals
1. Introduction . . .. .. . . . . .. . .. . .. .. .... .. . . .... ......... . . .. . 11.1 Fibonacci, Continued Fractions and the Golden Ratio . . . . . . 41.2 Fermat, Primes and Cyclotomy. . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Euler, Totients and Cryptography . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Gauss, Congruences and Diffraction. . . . . . . . . . . . . . . . . . . . . . 101.5 Galois, Fields and Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. The Natural Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 162.2 The Least Common Multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Planetary "Gears" 182.4 The Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Human Pitch Per ception. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Octaves, Temperament, Kilos and Decibels. . . . . . . . . . . . . . .. 212.7 Coprimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.8 Euclid's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3. Primes .. .. . .. . . .... . . . . ....... ... .. ... .. . .. . .. .. . . . .. . . . . 253.1 How Many Primes are There? . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 263.3 A Chinese Theorem in Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 A Formula for Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 283.5 Mersenne Primes 293.6 Repunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.7 Perfect Numb ers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343.8 Fermat Primes 353.9 Gauss and the Impossible Heptagon . . . . . . . . . . . . . . . . . . . . . . 36
4. The Prime Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1 A Probabilistic Argument " 384.2 The Prime-Counting Function 1l'(x) 404.3 David Hilbert and Large Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Coprime Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
XVI Contents
4.5 Primes in Progressions 484.6 Primeless Expanses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7 Squarefree and Coprime Integers . . . . . . . . . . . . . . . . . . . . . . . . . 514.8 Twin Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.9 Prime Triplets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 534.10 Prime Quadruplets and Quintuplets . . . . . . . . . . . . . . . . . . . . . . 544.11 Primes at Any Distance 554.12 Spacing Distribution Between Adjacent Primes. . . . . . . . . . .. 584.13 Goldb ach's Conjecture 584.14 Sum of Three Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
Part II. Some Simple Applications
5. Fractions: Continued, Egyptian and Farey . . . . . . . . . . . . . . .. 625.1 A Neglected Subject 625.2 Relations with Measure Theory " 665.3 Periodic Cont inued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 675.4 Elect rical Networks and Squared Squares. . . . . . . . . . . . . . . .. 705.5 Fibonacci Numb ers and the Golden Ratio. . . . . . . . . . . . . . . .. 715.6 Fibonacci, Rabbits and Computers. . . . . . . . . . . . . . . . . . . . . . . 755.7 Fibonacci and Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.8 Generalized Fibonacci and Lucas Numbers . . . . . . . . . . . . . . .. 785.9 Egypt ian Fractions, Inherit ance
and Some Unsolved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . 815.10 Farey Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
5.10.1 Farey Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.10.2 Locked Pallas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.11 Fibonacci and t he Problem of Bank Deposits. . . . . . . . . . . . .. 895.12 Error-Free Computing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Part III. Congruences and the Like
6. Linear Congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 956.1 Residues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2 Some Simple Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3 Powers and Congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7. Diophantine Equations " 1027.1 Relation with Congruences 1027.2 A Gaussian Trick 1037.3 Nonlinear Dioph antine Equ ations 1057.4 Tri angular Numb ers 1067.5 Pythagorean Numb ers 108
Contents XVII
7.6 Exponential Diophantine Equ ations 1097.7 Fermat's Last "Theorem" 1097.8 The Demise of a Conjecture by Euler 1117.9 A Nonlinear Dioph antine Equ ation in Physics
and the Geometry of Numb ers 1117.10 Norm al-Mod e Degeneracy in Room Acoustics
(A Number-Theoret ic Application) 1157.11 Waring's Problem 116
8. The Theorems of Fermat, Wilson and Euler 1188.1 Ferm at's Theorem 1188.2 Wilson's Theorem 1198.3 Euler's Theorem 1208.4 The Impossible Star of David 1218.5 Dirichlet and Linear Progression 123
Part IV. Cryptography and Divisors
9. Euler Trap Doors and Public-Key Encryption 1259.1 A Numerical Trap Door 1279.2 Digital Encryption 1289.3 Public-Key Encrypt ion 1299.4 A Simple Example 1319.5 Repeated Encryp tion 1329.6 Summary and Encryption Requirements 133
10. The Divisor Functions 13510.1 The Number of Divisors 13510.2 The Average of the Divisor Function 13810.3 The Geometric Mean of t he Divisors 13810.4 The Summatory Function of the Divisor Function 13910.5 The Generalized Divisor Functions 13910.6 The Average Value of Euler 's Function 140
11. The Prime Divisor Functions 14211.1 The Number of Different Prime Divisors 14211.2 The Distribution of w(n) 14611.3 The Numb er of Prime Divisors 14711.4 The Harmonic Mean of Q(n) 15011.5 Medians and Per centil es of Q(n) 15211.6 Implications for Public-Key Encryption 153
XVIII Contents
12. Certified Signatures 15412.1 A Story of Creative Financing 15412.2 Certifi ed Signature for Public-Key Encryption 154
13. Primitive Roots 15613.1 Ord ers 15613.2 Periods of Decimal and Binary Fractions 15913.3 A Primitive Proof of Wilson 's Theorem 16213.4 The Ind ex - A Number-Theoretic Logarithm 16213.5 Solution of Exponential Congruences 16313.6 Wh at is the Order Tm of an Integer m Modulo a Prime p? .. 16513.7 Index "Encrypt ion" 16613.8 A Fourier Property of Primitive Roots
and Concert Hall Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16613.9 More Spacious-Sounding Sound 16813.10 Galois Arrays for X-Ray Astronomy 17013.11 A Negative Property of the Ferm at Primes 171
14. Knapsack Encryption 17314.1 An Easy Knapsack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17314.2 A Hard Knapsack 174
Part V. Residues and Diffraction
15. Quadratic Residues 17715.1 Quadratic Congruences 17715.2 Euler's Criterion 17815.3 The Legendre Symbol " 17915.4 A Fourier Property of Legendre Sequences 18115.5 Gauss Sums 18115.6 Pretty Diffraction 18315.7 Quadratic Reciprocity 18315.8 A Fourier Property of Quadratic-Residue Sequences 18415.9 Spread Spectrum Communication 18615.10 Generalized Legendre Sequences Obtained
Through Complexification of the Euler Criterion 187
Part VI. Chinese and Other Fast Algorithms
16. The Chinese Remainder Theoremand Simultaneous Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 19016.1 Simultaneous Congruences 19016.2 The Sino-Representation: A Chinese Numb er System 191
Contents XIX
16.3 Applications of the Sino-Representation 19216.4 Discret e Fourier Transformation in Sino 19416.5 A Sino-Optical Fourier Transformer 19516.6 Generalized Sino-Represent at ion 19616.7 Fast Prime-Length Fourier Transform 197
17. Fast Transformation and Kronecker Products " 19917.1 A Fast Hadamard Transform 19917.2 The Basic Principle of the Fast Fourier Transforms 202
18. Quadratic Congruences 20318.1 Application of the Chinese Remainder Theorem (CRT) 203
Part VII. Pseudoprimes, Mobius Transform, and Partitions
19. Pseudoprimes, Poker and Remote Coin Tossing " 20519.1 P ulling Roots to Ferret Out Composites 20519.2 Factors from a Square Root 20619.3 Coin Tossing by Telephone 20819.4 Absolute and Strong Pseudoprimes 21019.5 Ferm at and Strong Pseudoprimes 21219.6 Deterministic Primality Test ing 21219.7 A Very Simple Factoring Algorithm 21419.8 Factoring with Elliptic Curves 21419.9 Quantum Factoring 215
20. The Mobius Function and the Mobius Transform 21620.1 The Mobius Transform and Its Inverse 21620.2 P roof of the Inversion Formula 21820.3 Second Inversion Formula 21920.4 Third Inversion Formula 21920.5 Fourth Inversion Formula 22020.6 Riemann's Hypothesis
and the Disproof of the Mertens Conjecture . . . . . . . . . . . . . . . 22020.7 Dirichlet Series and th e Mobius Function 221
21. Generating Functions and Partitions 22421.1 Gener ating Fun ctions 22421.2 Partitions of Integers 22621.3 Generating Functions of Partitions 22721.4 Restricted Partitions 228
XX Contents
Part VIII. Cyclotomy and Polynomials
22. Cyclotomic Polynomials 23222.1 How to Divide a Circle into Equ al Parts 23222.2 Gauss 's Great Insight 23522.3 Factoring in Different Fields 23922.4 Cyclotomy in the Complex Plane 23922.5 How to Divide a Circle with Compass and Straightedge 240
22.5.1 Rational Factors of zN - 1 24222.6 An Alternative Rational Factorization 24322.7 Relation Between Rational Factors and Complex Roots 24422.8 How to Calculate with Cyclotomic Polynomials 245
23. Linear Systems and Polynomials 24723.1 Impulse Responses 24723.2 Time-Discrete Systems and the z Transform 24823.3 Discrete Convolution 24823.4 Cyclotomic Polynomials and z Transform 249
24 . Polynomial Theory 25024.1 Some Basic Facts of Polynomial Life 25024.2 Polynomial Residues 25124.3 Chinese Remainders for Polynomials 25224.4 Euclid's Algorithm for Polynomials 253
Part IX. Galois Fields and More Applications
25 . Galois Fields 25625.1 Prime Order 25625.2 Prime Power Order 25625.3 Generation of GF(24 ) • • ..• . . . . . .. . . .... • ••. . . . .. . .. . . . . 25825.4 How Many Primitive Elements? 26025.5 Recursive Relations 26025.6 How to Calculate in GF(pffi) 26225.7 Zech Logarithm, Doppler Radar
and Optimum Ambiguity Functions . . . . . . . . . . . . . . . . . . . . . . 26325.8 A Uniqu e Phase-Array Based on the Zech Logarithm 26625.9 Spread-Spectrum Communication and Zech Logarithms 268
26. Spectral Properties of Galois Sequences 26926.1 Circular Correlation 26926.2 Application to Error-Correcting Codes
and Speech Recognition 271
Contents XXI
26.3 Applic ation to Precision Measurements 27326.4 Concert Hall Measurement s 27426.5 The Fourth Effect of General Relativity 27526.6 Toward Better Concert Hall Acoustics 27626.7 Higher-Dimensional Diffusors 28126.8 Active Array Applications 282
27 . Random Number Generators 28327.1 Pseudorandom Galois Sequences 28427.2 Randomness from Congruences 28527.3 "Cont inuous" Distributions 28627.4 Four Ways to Generate a Gaussian Variable 28727.5 Pseudorandom Sequences in Cryptography 288
28 . Waveforms and Radiation Patterns .. " . . " 28928.1 Special Phases 29028.2 The Rudin-Shapiro Polynomials 29228.3 Gauss Sums and Peak Factors 29328.4 Galois Sequences and the Smallest Peak Factors 29528.5 Minimum Redundancy Antennas 29728.6 Golomb Rulers 299
29. Number Theory, Randomness and "Ar t " 30129.1 Number Theory and Graphi c Design 30129.2 The Primes of Gauss and Eisenstein 30329.3 Galois Fields and Imp ossible Necklaces 30429.4 "Baroque" Integers 308
Part X . Self-Similarity, Fractals and Art
30. Self-Similarity, Fractals, Deterministic Chaosand a New State of Matter " .. " 31130.1 Fibonacci, Noble Numb ers and a New State of Mat te r 31530.2 Cantor Sets, Fractals and a Musical Paradox 32030.3 The Twin Dragon:
A Fractal from a Complex Number System 32530.4 St ati st ical Fractals 32730.5 Some Crazy Mappings 32930.6 The Logistic Parabol a and Strange Attractors 33230.7 Conclusion 335
XXII Contents
Glossary of Symbols 336
References 339
Name Index 351
Subject Index 355
