Recent Titles in This Series

47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993)

46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991)

45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992)

44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991)

43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado,

August 1989) 41 R. W. Brockett, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January

1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind

the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January

1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987)

36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986)

35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985)

34 Michael Anshel and William Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984)

33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985)

32 R. W. McKelvey, editor, Environmental and natural resource mathematics (Eugene, Oregon, August 1984)

31 B. Gopinath, editor, Computer communications (Denver, Colorado, January 1983)

30 Simon A. Levin, editor, Population biology (Albany, New York, August 1983) 29 R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied

cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981)

28 R. Gnanadesikan, editor, Statistical data analysis (Toronto, Ontario, August 1982) 27 L. A. Shepp, editor, Computed tomography (Cincinnati, Ohio, January 1982)

26 S. A. Burr, editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981) 25 S. I. Gass, editor, Operations research: mathematics and models (Duluth, Minnesota,

August 1979)

24 W. F. Lucas, editor, Game theory and its applications (Biloxi, Mississippi, January 1979)

23 R. V. Hogg, editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980)

22 G. H. Golub and J. Oliger, editors, Numerical analysis (Atlanta, Georgia, January 1978) 21 P. D. Lax, editor, Mathematical aspects of production and distribution of energy (San

Antonio, Texas, January 1976)

20 J. P. LaSalle, editor, The influence of computing on mathematical research and education (University of Montana, August 1973)

19 J. T. Schwartz, editor, Mathematical aspects of computer science (New York City, April 1966)

18 H. Grad, editor, Magneto-fluid and plasma dynamics (New York City, April 1965) {Continued in the back of this publication)

http://dx.doi.org/10.1090/psapm/046http://dx.doi.org/10.1090/psapm/046

The Unreasonable Effectiveness of Number Theory

AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures

published as a subseries of Proceedings of Sjmiposia in Applied Mathematics

Proceedings of Symposia in

APPLIED MATHEMATICS

Volume 46

The Unreasonable Effectiveness of Number Theory

American Mathematical Society Short Course August 6-7, 1991 Orono, Maine

Stefan A. Burr, Editor

'TPHTOI

* ^ ^ ijl American Mathematical Society 3 Providence, Rhode Island

LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE

THE UNREASONABLE EFFECTIVENESS OF NUMBER THEORY HELD IN ORONO, MAINE

AUGUST 6-7, 1991

The AMS Short Course Series is sponsored by the Society's Program Committee for National Meetings. The series is under the direction of the

Short Course Subcommittee of the Program Committee for National Meetings.

1991 Mathematics Subject Classification. Primary 11-06, 11K45, 11T71, 11Z50.

Library of Congress Cataloging-in-Publication Data The unreasonable effectiveness of number theory / Stefan A. Burr, editor; George E. Andrews ... [et al.].

p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 46. AMS short course lecture notes)

Course held in Orono, Maine, Aug. 6-7, 1991. Includes index. ISBN 0-8218-5501-8 (hard bound; acid free) 1. Number theory-Congresses. I. Burr, Stefan A. (Stefan Andrus), 1940- . II. Andrews,

George E., 1938- . III. Series: Proceedings of symposia in applied mathematics; v. 46. IV. Series: Proceedings of symposia in applied mathematics. AMS short course lecture notes. QA241.U67 1992 92-24328 512'.7-dc20 CIP

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 an article 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.

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© Copyright 1992 by the American Mathematical Society. All rights reserved. Printed in the United States of America.

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10 9 8 7 6 5 4 3 01 00 99 98 97

Table of Contents

Preface

The Unreasonable Effectiveness of Number Theory in Physics, Communication and Music MANFRED R. SCHROEDER

The Reasonable and Unreasonable Effectiveness of Number Theory in Statistical Mechanics GEORGE E. ANDREWS

Number Theory and Dynamical Systems J. C. LAGARIAS

The Mathematics of Random Number Generators GEORGE MARSAGLIA

Cyclotomy and Cyclic Codes VERA PLESS

Number Theory in Computer Graphics M. DOUGLAS MCILROY

Index

Preface

In August 1991, the American Mathematical Society sponsored a Short Course at the summer meeting in Orono, Maine, entitled "The Unreasonable Effectiveness of Number Theory". Two years earlier, another Short Course was held on "Cryptology and Computational Number Theory", which em­phasized cryptologic applications. Therefore, the Short Course in Orono concentrated on the great breadth of applications outside cryptology. This volume is based on the lectures given at that Short Course.

Number theory is one of the oldest and noblest branches of mathematics; indeed, it was already ancient in the time of Euclid. In fact, for almost all of its history it has seemed to be among the purest branches of mathematics. It is only within the last few decades that a large number of applications have been encountered, at least by the mathematical community. The applications to cryptology are now famous; but it is not as well known that number the­ory has found an enormous number and variety of real-world applications in many different fields. Indeed, the standard Mathematics Subject Classifi­cation includes several codes devoted entirely or heavily to applications of number theory.

What are the sources of these applications? The largest impetus has been, not surprisingly, the computer, with its digital and numerical nature. As with cryptology, the computer has been a driving force in the development of algebraic coding theory, random number generation, raster graphics, com­puter arithmetic, fast transforms, and many other areas. Perhaps surpris­ingly, physics, in spite of its tradition of being continuous rather than dis­crete, is another rich source of applications. Here, many of these fall into two categories: periodic phenomena and special functions. Here are just a few spatially or temporally periodic phenomena in physics: acoustics, diffrac­tion, antenna design, dynamical systems, resonances of astronomical bodies, and crystal (and quasicrystal) structure. Number theory also has an affin­ity with special functions, for instance those arising in statistical mechanics. Here, the connection often involves the fact that many such special functions are the generating functions of sequences arising in additive (and sometimes multiplicative) number theory. It should be mentioned that there are many applications of number theory that do not come from computers or physics;

xii PREFACE

for instance in chemistry, engineering, biology, and the arts. This book will introduce the reader to many of the above fields, giving some idea of the riches that exist. Many riches remain unmentioned, but the subject is too large to encompass in a single volume.

The title of the Short Course and of this book are a salute to the famous phrase of Wigner, who marveled at "the unreasonable effectiveness of mathe­matics" in the real world. Until recently, few people would have applied this phrase to number theory, but the authors and I hope that this volume will help to further the ferment that exists in the applications of number theory.

Stefan A. Burr

Index

acoustics, 8 action-angle variables, 50 additive continued fraction, 38, 41 additive number theory, 21 analytically conjugate, 55 aperiodic tilings, 63 Arnol'd tongues, 57 attracting fixed point, 54

Baxter's identity, 30 bending of light, 16 binary operation, 77 binary quadratic residue codes, 96 block-substitution, 66 Born, Max, 2 Bruckstein, A. M., 114, 119

canonical transformation, 46 cantori, 52 carry bit, 79, 80 chain codes, 113 chemical elements, 3 Chinese remainder, 10 Circle of Fifths, 5 completely integrable, 47, 48 computer, 1 computer graphics, 105 concert halls, 5 conserved quantity, 46 continued fraction, 4, 114, 115 corner transfer matrices, 24, 26 cryptography, 10 cut-and-project method, 39, 45, 63 cutting sequence, 114 cyclic code, 93

delta functions, 62 diffraction theory, 6 digital circle, 108 digital communication, 1 digital curve, 106, 113

digital ellipse, 112 digital line, 113-118 digital segment, 113, 116-119 Diophantine approximation, 36, 105 Diophantine condition, 55 directed graph, 81, 82 directed graph, component, 81 discrete mathematics, 1 divisible code, 102 Dorst, L., 119 doubly even, 93

Eddington, 16 eigenvalues, 26 Einstein, 2, 15 elementary particles, 1 ellipsoid method, 119 elliptic function, 24, 26 equivalent, 101 ergodic theory, 36 error-correcting code over GF{q), 91 error-correcting codes, 12 Euler's theorem, 11 Euler's totient, 118 Euler's totient function, 10 even-like duadic codes, 96 exclusive-or, 78, 79, 89 extended code, 92

Farey fan, 117, 118 Farey neighbors, 41 Farey series, 117, 118 Farey shift map, 38, 41, 43 Farey sum, 41 Farey tree, 41 Fermat prime, 3, 10 finite fields, 12 Forchhammer, S., 119 Fourier transform, 6, 1 5 Freeman approximation, 106-108 fundamental sequence, 75

124 INDEX

G-completely integrable, 50 Galois arrays, 12 Galois fields, 12 Galois sequences, 12 Gauss map, 39 Gauss measure, 40 general linear group, 116 general relativity, 15 generalized multipliers, 101 generator matrix, 91 genetic code, 3 geodesic flow, 49 gravitational field, 16 gravitational red shift, 16 group of a design, 100 group of the code, 93

Hamiltonian flows, 46 Hamiltonian function, 46 Hamming Codes, 13 Hard Hexagon model, 21, 24, 26 Hausdorff dimension, 44 Heine's transformation, 24 Heisenberg, Werner, 2 Hilbert, David, 2, 15 hologram, 3

idempotent generator, 94 incidence matrix, 100 indifferent fixed point, 54 inflation, 64 integer, 1 integrals of the motion, 47 intermediate convergents, 41 interplanetary distance measurements, 15 invariant circles, 52 irreducible polynomial, 12 isotopes, 3

KAM theory, 50 Kirchhoff approximation, 6 Knuth, Donald E., 74, 90

"linear code", 13 Liouville's theorem, 46

Markoff spectrum, 114-115 Marsaglia, George, 74, 75, 78, 79, 90 "matrix" mechanics, 2 matter waves, 4 mediant, 41 Mercury, 16 Metric entropy, 44 minimum weight of the code, 91 Minkowski, Hermann, 2 Minkowski's ?-function, 44 mode-locking, 38, 56 modulated crystals, 63

Moessbauer effect, 16 monotone twist map. 57 multipliers, 95 music, 5

Newton's method, 119 nonparametric statistics, 22 nonresonant, 51 number theory, 1

odd-like duadic codes, 96 orchard, 119 order of the design, 99 ordinary continued fraction expansion, 39

paper folding sequences, 65 parity-check matrix, 92 partial quotients, 39 partial remainders, 39 partition function, 25 partitions, 21, 22, 23, 28, 29 Pell equation, 111 perfect fifth, 4 perihelion motion, 16 phase grating, 8 physics, 1 Pierce, J. R„ 5 Pisot number, 66 Poincare-Birkhoff twist theorem, 53 Poisson bracket, 46 "polynomial-time" algorithm, 12 position estimate, 119 primitive root, 8 projective plane, 100 public-key cryptography, 10 Pythagoras, 3, 4

g-cyclotomic cosets, 94 quadratic residue, 6 quantum mechanics, 1 quasicrystals, 59

Ramanujan's Lost Notebook, 29 random number generator, add-with-carry,

80-83, 85-87 random number generator, combination,

75, 89 random number generator, congruential,

73-75, 89 random number generator, effective period,

75 random number generator, examples of, 89 random number generator, extended con­

gruential, 77, 79, 89 random number generator, full (maximal)

period, 76, 78, 79 random number generator, lagged Fi­

bonacci, 74, 76-81, 89

INDEX 125

random number generator, period, 75, 85-89

random number generator, seeds, 79, 81, 88, 89

random number generator, shift register, 74, 75-76

random number generator, subtract-with-borrow, 83, 87-89

random number generators, 73-90 Rayleigh, 2 reflection phase-gratings, 5 renormalization, 45, 58 repelling fixed point, 54 residue, 53 Ripley, Brian D., 90 Rogers-Ramanujan identities, 21, 28 rotation intervals, 59 rotation number, 52, 57

"secure" communication, 3 self-dual, 92 self-generating sequence, 66 self-orthogonal, 92 Simple Harmonic Oscillator, 48 Simplex Code, 13 sine circle map, 56 skew-product, 40

slowing of electro-magnetic radiation, 16 small divisors, 38 stability, 38 standard map, 52 statistical mechanics, 21, 24, 33 substitution rules, 63 substitution sequences, 39 superior conjunction, 16 symbolic dynamics, 35, 42 symmetric (v,k,X) design, 99 symplectic form, 46

tempered distributions, 62 Thue-Morse sequence, 65 Toda Lattice, 49 topological dynamics, 35 topological entropy, 45 trap-door function, 11 Tsay, L. H., 78, 90

Venus, 16

wave interference, 3 weight of a vector, 91 wt(x), 91

Zaman, Arif, 79, 90

Recent Titles in This Series (Continued from the front of this publication)

17 R. Finn, editor, Applications of nonlinear partial differential equations in mathematical physics (New York City, April 1964)

16 R. Bellman, editor, Stochastic processes in mathematical physics and engineering (New York City, April 1963)

15 N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962)

14 R. Bellman, editor, Mathematical problems in the biological sciences (New York City, April 1961)

13 R. Bellman, G. Birkhoff, and C. C. Lin, editors, Hydrodynamic instability (New York City, April 1960)

12 R. Jakobson, editor, Structure of language and its mathematical aspects (New York City, April 1960)

11 G. Birkhoff and E. P. Wigner, editors, Nuclear reactor theory (New York City, April 1959)

10 R. Bellman and M. Hall, Jr., editors, Combinatorial analysis (New York University, April 1957)

9 G. Birkhoff and R. E. Langer, editors, Orbit theory (Columbia University, April 1958) 8 L. M. Graves, editor, Calculus of variations and its applications (University of Chicago,

April 1956) 7 L. A. MacColl, editor, Applied probability (Polytechnic Institute of Brooklyn, April

1955) 6 J. H. Curtiss, editor, Numerical analysis (Santa Monica City College, August 1953) 5 A. E. Heins, editor, Wave motion and vibration theory (Carnegie Institute of

Technology, June 1952) 4 M. H. Martin, editor, Fluid dynamics (University of Maryland, June 1951) 3 R. V. Churchill, editor, Elasticity (University of Michigan, June 1949) 2 A. H. Taub, editor, Electromagnetic theory (Massachusetts Institute of Technology, July

1948) 1 E. Reissner, editor, Non-linear problems in mechanics of continua (Brown University,

August 1947)