http://dx.doi.org/10.1090/psapm/046
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)
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
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Table of Contents
Preface
The Unreasonable Effectiveness of Number Theory in Physics,
Communication and Music
M A N F R E D R. SCHROEDER
The Reasonable and Unreasonable Effectiveness of Number Theory
in Statistical Mechanics
G E O R G E E. ANDREWS
Number Theory and Dynamical Systems
J. C. LAGARIAS
The Mathematics of Random Number Generators
G E O R G E MARSAGLIA
Cyclotomy and Cyclic Codes
VERA PLESS
Number Theory in Computer Graphics
M. DOUGLAS M C I L R O Y
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
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
INDEX
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 congruential, 77, 79, 89
random number generator, full (maximal)
period, 76, 78, 79
random number generator, lagged Fibonacci, 74, 76-81, 89
125
INDEX
random number generator, period, 75, 8589
random number generator, seeds, 79, 81,
88, 89
random number generator, shift register,
74, 75-76
random number generator, subtract-withborrow, 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,
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11 G. Birkhoff and E. P. Wigner, editors, Nuclear reactor theory (New York City, April
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10 R. Bellman and M. Hall, Jr., editors, Combinatorial analysis (New York University,
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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,
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7 L. A. MacColl, editor, Applied probability (Polytechnic Institute of Brooklyn, April
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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
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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
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