INTRODUCTION TO NUMBER THEORY

DANIEL E. FLATH

AMS CHELSEA PUBLISHING

IntroductIon to number theory

IntroductIon to number theory

danIel e. Flath

AMS CHELSEA PUBLISHING

10.1090/chel/384.H

2010 Mathematics Subject Classification. Primary 11-01.

For additional information and updates on this book, visitwww.ams.org/bookpages/chel-384

Library of Congress Cataloging-in-Publication Data

Names: Flath, Daniel E., author.

Title: Introduction to number theory / Daniel E. Flath.

Other titles: Number theory

Description: [2018 edition]. | Providence, Rhode Island : American Mathematical Society, 2018.| Series: AMS Chelsea Publishing [series] ; 384 | Originally published: New York : Wiley, 1989. |Includes bibliographical references and indexes.

Identifiers: LCCN 2018014214 | ISBN 9781470446949 (alk. paper)Subjects: LCSH: Number theory. | AMS: Number theory – Instructional exposition (textbooks,tutorial papers, etc.). msc

Classification: LCC QA241 .F59 2018 | DDC 512.7–dc23LC record available at https://lccn.loc.gov/2018014214

Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingfor them, are permitted to make fair use of the material, such as to copy select pages for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for permissionto reuse portions of AMS publication content are handled by the Copyright Clearance Center. Formore information, please visit www.ams.org/publications/pubpermissions.

Send requests for translation rights and licensed reprints to reprint-permission@ams.org.

c© 1989 held by Daniel E. Flath. All rights reserved.Reprinted by the American Mathematical Society, 2018

Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at https://www.ams.org/

10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18

www.ams.org/bookpages/chel-384www.ams.org/publications/pubpermissionsreprint-permission@ams.orghttps://www.ams.org/

To Laura

Preface

When I came to Singapore I was given a fourth year undergraduate Honours Number Theory course. I decided to teach Gauss’s immortal Disquisitiones Arithmeticae. This book is the result.

On historical and mathematical grounds alike number theory has earned a place in the curriculum of every mathematics student. This is a textbook for an advanced undergraduate or beginning graduate core course in the subject. Such a course should stick pretty close to the naive questions, which in number theory concern prime numbers and Diophantine equations. The emphasis in this book is on Diophantine equations, especially quadratic equations in two variables.

My own conscious interest in Diophantine equations goes back to a long winter’s night in a St. Louis basement in 1962 when my father and I tried to solve the notorious problem of the monkey and the coconuts as presented by Martin Gardner. No one told me then that Diophantine equations belong to a subject called “ number theory,” and I found little help in the public library. I needed a teacher trained in number theory. It pleases me that several of my students of Gauss are now teaching in the schools. I might particularly mention Mr. Lee Ah Huat with whom I discovered Gauss’s first proof of the law of quadratic reciprocity.

This book is closely based on lectures I gave to able groups of students during three consecutive years at the National University of Singapore. I thank the students for constantly demanding “ the notes,” which was how the text began. I tried during the writing always to keep my students in mind, always to remember that I was writing a textbook. I have sought to avoid the twin traps of doing algebra to the exclusion of number theory and of doing only trivial number theory. I take it for granted that the material I have chosen is interesting. My supreme stylistic goal is clarity.

By the time this book is published I shall have gone on from Singapore. Singapore has been part of my life for three and a half years, and I shall miss it. I have many friends here. I wish them all well.

Viii PREFACE

Singapore August 1987

Dan Flath

Acknowledgment

I am very grateful to Jean-Pierre Serre and the Singapore Mathematical Society for permission to reprint the article “ A = b2 - Aac” which was first published in the Mathematical Medley.

Daniel E. Flath

ix

Contents

1. Prime Numbers and Unique Factorization 1

1. Introduction, 12. Prime Numbers, 23. Unique Factorization, 64. 4- #2 ^ 2 * *' +QrX r = n, 115. The Distribution of the Primes, 17

2. Sums of Two Squares 24

1. Introduction, 242. Integers mod m, 253. Applications of Lemma 1.4, 324. Gaussian Integers, 345. Farey Sequences, 396. Minkowski’s Theorem, 477. Method of Descent, 508. Reduction of Positive Definite Binary Quadratic Forms, 55

3. Quadratic Reciprocity 63

1. Introduction, 632. Composite Moduli, 653. The Legendre Symbol, 704. The First Proof, 775. The Gauss Lemma, 80

xi

xii CONTENTS

6. Gauss Sums, 837. The Ring Z[e2”i/n], 898. The Jacobi Symbol, 929. The Kronecker Symbol, 95

10. Binary Quadratic Forms, 99

4. Indefinite Forms 104

1. Introduction, 1042. The Square Root of 2, 1053. The Pell Equation, 1084. a X 2 + b X Y + c Y 2 = m, 1185. Automorphisms, 1246. Reduction of Indefinite Forms, 1287. Continued Fractions, 1358. Reduction (II), 1399. Automorphisms (II), 142

5. The Oass Group and Genera 147

1. Introduction, 1472. The Class Group, 1483. The Genus Group, 1564. What Gauss Did, 1615. Counting Ambiguous Classes, 1646. The Ternary Form Y 2 - XZ, 1687. The Duplication Theorem, 1728. Sums of Three Squares, 175

Appendix A. A = b1 — 4 ac by Jean-Pierre Serre 180

Appendix B. Tables 190

Bibliography 204

Subject Index 207

Notation Index 211

Bibliography

Number theory has been blessed with many excellent books. This bibliography is a list of those that were available to me during the writing of this book. I have drawn most heavily from [5], [12], [13], [14], [31], and [33].

The reader who would learn more about rational and integral quadratic forms must begin by studying the fields of />-adic numbers. There are fine introductions in [4] and [27]. His ultimate goal must be a close study of [5], which is magnificent.

A proof of Dirichlet’s Theorem on Primes in Arithmetic Progressions can be found in [27]. For a superb introduction to the distribution of prime numbers, see [28].

[1] Auslander, L. and Tolimieri, R., Ring structure and the Fourier transform, TheMathematical Intelligencer 7, 49-52 (1985).

[2] Bachmann, P., Niedere Zahlentheorie y Chelsea, New York, 1968.[3] Baker, A., A Concise Introduction to the Theory o f Numbers, Cambridge University

Press, Cambridge, 1984.[4] Borevich, Z. I. and Shafarevich, I. R., Number Theory, Academic, New York,

1966.[5] Cassels, J. W. S., Rational Quadratic Forms, Academic, London, 1978.[6] Cohn, H., A Second Course in Number Theory, Wiley, New York, 1962.[7] Davenport, H., The Higher Arithmetic, 5th ed., Cambridge University Press,

Cambridge, 1982.[8] Dickson, L. E., Introduction to the Theory o f Numbers, The University of Chicago

Press, Chicago, 1929.[9] Dirichlet, P. G. L., Vorlesungen uber Zahlentheorie, herausgegeben von R.

Dedekind, 4th ed., Chelsea, New York, 1968.[10] Edwards, H. M., Fermat's Last Theorem, Springer, New York, 1977.

204

BIBLIOGRAPHY 205

[11] Frei, G., Leonhard Euler’s Convenient Numbers, The Mathematical Intelligencer, 7, 55-58, 64 (1985).

[12] Gauss, C. F., Disquisitiones Arithmeticae, Springer, New York, 1986.[13] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 5th

ed., Oxford University Press, Oxford, 1979.[14] Hua, L. K., Introduction to Number Theory, Springer, New York, 1982.[15] Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory,

Springer, New York, 1982.[16] Jones, B. W., The Arithmetic Theory of Quadratic Forms, The Mathematical

Association of America, Providence, R.I., 1950.[17] Knuth, D. E., The Art of Computer Programming, Vol. 2, 2nd ed., Addison-Wesley,

Reading, Mass., 1981.[18] Landau, E., Elementary Number Theory, Chelsea, New York, 1958.[19] Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, New

York, 1953.[20] LeVeque, W. J., Fundamentals of Number Theory, Addison-Wesley, Reading,

Mass., 1977.[21] LeVeque, W. J., Topics in Number Theory, Vol. II, Addison-Wesley, Reading,

Mass., 1956.[22] Mathews, G. B., Theory of Numbers, Part I, Deighton, Bell and Co., Cambridge,

1892.[23] Mordell, L. J., Diophantine Equations, Academic, London, 1969.[24] Nagell, T., Introduction to Number Theory, 2nd ed., Chelsea, New York, 1981.[25] Narkiewicz, W., Number Theory, World Scientific, Singapore, 1983.[26] Niven, I. and Zuckerman, H. S., An Introduction to the Theory of Numbers, 4th ed.,

Wiley, New York, 1980.[27] Serre, J.-P., A Course in Arithmetic, Springer, New York, 1973.[28] Shapiro, H. N., Introduction to the Theory of Numbers, Wiley, New York, 1983.[29] Sierpinski, W., Elementary Theory of Numbers, Panstwowe Wydawnictwo

Naukowe, Warsaw, 1964.[30] Stark, H. M., An Introduction to Number Theory, MIT Press, Cambridge, Mass.,

1978.[31] Venkov, B. A., Elementary Number Theory, Wolters-Noordhoff Publishing,

Groningen, 1970.[32] Vinogradov, I. M., Elements o f Number Theory, Dover, London, 1954.[33] Weil, A., Number Theory: An Approach through History, Birkhauser, Boston, 1983.[34] Weil, A., Number Theory for Beginners, Springer, New York, 1979.

Subject Index

Algebraic number theory, 64, 89 Ambiguous class, see Class group Ambiguous form, see Quadratic forms, binary Automorphism, see Quadratic formsBaker, A., 187Binomial coefficients, 19, 20, 30 Birch and Swinnerton-Dyer conjecture, 188 Box principle, 48. See also Pigeon-hole

principleCharacter, group, 88 Chebyshev, P. L., 1, 3,17, 19 Chinese remainder theorem, 64, 65-66 Class group, 102, 147, 152-156, 160,183. See

also Class number; Quadratic forms, binary

ambiguous class, 159, 160, 164-168 composition:

of classes, 151-153 of forms, 149, 154

concordant forms, 149, 150,155 principal class, 152, 154

Class number, 133, 183-189. See also Class group

equals one, 155-156, 184-185 finiteness, 58, 61-62, 129, 152, 181

Composite integer, 2, 3, 94 witness, 94

Composition, see Class group; Product identities

Concordant forms, see Class group Congruences, 25, 26. See also Integers mod m

biquadratic x4 = a, 34, 95, 103

composite modulus 65-67

linear, 66, 69, 127, 155 quadratic, see also Legendre symbol

x 1 a - 1 , 25, 27-28, 30, 178 x 2 a ± 2 , 31, 53-54, 76 x 2 a - 3 , 31, 38 X 2 S 5, 31x 2 a a, 28, 63-64, 67-69, 70, 95, 97-98,

99, 102x 2 + y 2 a - 1 , 52 x 2 + y 2 a a, 55 others, 95, 175-177

Conjugation in quadratic ring, 111, 117 Continued fractions, 10, 135, 138

nearly simple, 135-136, 140 Convenient numbers, see Genus group Convex set, 50 Cyclic groups, 29, 31, 131 Cyclotomic ring, 89Decimal fractions, 31 Dedekind domain, 89 Descent, 25, 50-52, 105

infinite, 52 Deuring, M., 186Diophantine approximation, see Rational

approximationDiophantine equations, vii. See also Sum of

squares; Pell equation linear, 1, 11-16, 119,124

positive solutions, 16, 17 quadratic:

x 2 = 2 y 2, 52, 105x 2 + y 2 —n, see Sum of squares, two x 2 ± 2y 2 - n, 33-34, 50, 53-54 x 2 + 3y 2 = n, 38, 62 x 2 + xv + y 2 = 11, 38, 60, 62

207

208 SUBJECT INDEX

Diophantine equations, vii. (Continued) x 2 + 5y 2 - /i, 102, 148, 154, 160 x 2 + 36>>2 - p , 103, 161 x 2 + 64y 2 =* p , 95, 161 y 2 - 2*2 + 1, 105-108 x 2 - dy2 — ±1, see Pell equation x 2 — dy2 — m, 123, 124 ax2 + bxy + cy2 =» m, 104-105, 118-123,

128,134. See also Prime numbers, representation by quadratic forms

x 2 + y 2 — z2, 54 others:

y - X3 + k , 34, 39X 2 + 3 y 2 - r 3, 39X3+ y 3 - r 3, 55X3+ p y 5 + />2z3 - 0, 53X3+ dy 3 + d 2z 3 - id x y z = 1, 118y 2 - X3 - x, 55X4 + - z4, 55X4 + / = z2, 54X4 -1- / = z4, 54

Dirichlet, P. G. L., primes in arithmeticprogressions, 4, 5, 23, 99, 158-159, 162, 172, 204

3k + 2, 54k 4- ft, 4, 22-23, 24, 325k - 1, 766* + 5, 58* + ft, 34, 7610* + 1, 3210* + 9, 76p"k + 1, 32

Discriminant, 95,180. See a/so Quadratic forms, binary

fundamental, 183,184 Divisibility, 2, 25

divisor a product, 10 of factors in product, 8, 9

Division algorithm, 6, 28, 35 Divisors:

common, 6greatest common (GCD), 1, 6, 7, 9, 12, 13,

35-36number, 10 sum, 10, 11

Duplication theorem, see Genus group

Egyptian fraction, 47Elementary divisors, 16Elliptic curve, 187-188Equivalence of forms, see Quadratic formsEratosthenes, 3Euclid, 2, 17, 34

Euclidean algorithm, 6, 36,119 Euclidean domain, 36 Euclid’s lemma, 8Euler, L., 17, 22, 23, 52, 63, 73, 161,185 Euler product, 17-18, 22-23

Factorization, 119 of factorials, 20 uniqueness, 36, 89

Gaussian integers, 25, 34-36 integers, 1, 8-9, 18 other rings, 38, 39

Farey sequences, 39-45, 47 Father, viiFermat, P., 5, 24, 50, 51

last theorem, 54, 55, 89 little theorem, 26-27, 30

Fibonacci numbers, 9, 118 Fourier transform, 84-89 Frobenius, G., 185Fundamental theorem of arithmetic, see

Factorization, uniqueness Fundamental unit, see Order, units in

Gardner, Martin, viiGauss, C. F., 1, 17, 34, 63-64, 72, 77, 80, 83,

102,129,147,149,157,162,163,178,180, 181, 183, 184

Gaussian form, see Quadratic forms, binary Gaussian integers, 34-38

generalizations, 89,111-112. See also Order Gauss lemma, 64, 80-83 Gauss sums, 64, 83, 89, 90-91 Gauss symbol, see Genus group Genus group, 102,147-148,157-158. See also

Class groupconvenient numbers, 161 duplication theorem, 159,160,162,168,172,

175Gauss symbol, 157 principal genus, 157, 160,175

Geometry of numbers, see Minkowski’s theorem

Goldfeld, D. M., 187, 188 Greatest common divisor, see Divisors, greatest

common Gronwall, 186 Gross, B. H., 188

Hadamard, J., 17 Heegner, K., 186, 187 Heilbronn, H., 186 Hensel’s lemma, 65, 69

SUBJECT INDEX 209

Identity, see Product identities Infinite descent, see Descent, infinite Integers, see also Factorization; Prime numbers

nonsquare, 99, 109 relatively prime, 7 squarefree, 11

Integers mod m, 25, 26. See also Congruences invertible, 26, 30 primitive roots, 29-32, 54, 74, 75 roots of polynomials, 28, 30, 67, 69

Irrational numbers, 11,105,109, 111, 116

Jacobi symbol, 92-93

Kronecker symbol, 64, 95-99, 100, 157-158, 160

Lagrange, J. L., 52, 74 Landau, E., 23, 186 Lattices:

integer, 40-43 in R2, 47

fundamental parallelogram, 47 Least common multiple, 11 Lee Ah Huat, vii Legendre, A. M., 17Legendre symbol, 64, 70-73, 80, 92, 93. See

also Quadratic reciprocity evaluation, see also Congruences, quadratic

( - 1 / P \ 71, 81 (2 /p ), 71-72, 81, 91, 93 (5//>), 74-75

L series, 23, 187

Mestre, J.-F., 188 Minkowski’s theorem, 25, 47-50 Modular forms, 187 Modular functions, 185, 186 Module of binary form, See Quadratic forms,

binaryMontgomery, H. L., 187,188-189

Negative Pell equation, see Pell equation, negative

Neighbor of binary form, see Reduction, binary forms

Newton’s method, see Hensel’s lemma Nonsquare integers, see Integers, nonsquare Norm in quadratic ring, 34,111-112,117,120

Order, 112,117, 122, 183 square discriminants, 113, 117

units in, 105, 112-114, 120. See also Pell equation

fundamental unit, 114,117

Partial tractions, 16Pell equation, 104-117,119,124-126, 145

and automorphisms of forms, 105,124-126, 142, 145

negative, 109, 116, 127,168 Pell form, 108and units in orders, 105,113-114,117

Pell form, see Pell equation, Pell form Perfect numbers, 11Period of quadratic form, see Reduction,

binary forms Pi, 23, 45Pigeon-hole principle, 110. See also Box

principlePrimality tests, 3, 65, 93-94 Prime numbers, vii, 1, 2, 93. See also

Factorizationin arithmetic progressions, see Dirichlet distribution, 1, 3, 17, 19, 21, 204 generated by polynomial, 6 infinitely many, 1, 2, 5,17,18 largest known, 5representation by quadratic forms, 56, 59,

63, 64, 99-103, 148, 159-160 size of nth, 21sum of reciprocals, 17-19, 22

Prime number theorem, 17 Primitive roots, see Integers mod m, primitive

rootsPrincipal class, see Class group Principal form, see Quadratic forms, binary Principal genus, see Genus group Product identities, 33, 52,102, 106, 112, 148,

154Pythagoras, 105Pythagorean triples, see Diophantine

equations, x 2 4- y 2 *» z 2

Quadratic field, 111, 183,185, 186 Quadratic forms, 204. See also Diophantine

equations; Reduction binary, 5, 55. See also Class group; Genus

groupambiguous, 165, 168

special, 165, 168 automorphisms, 105, 124

improper, 125, 126-127, 134 proper, 105,125-126,142,144-145,166.

See also Pell equation

210 SUBJECT INDEX

Quadratic forms (Continued)discriminant, 55, 60. See also Discriminant equivalence, 57, 61, 100-102

proper, 57, 61-62, 102, 105, 128, 147, 151-153, 181

gaussian, 152 integral, 55 module, 119Pell form, see Pell equation, Pell form positive definite, 55, 61 primitive, 101, 183 principal, 152representation of integers by, 55,156. See

also Prime numbers; Diophantine equations

proper, 102, 124, 128, 134 roots, 139-140, 143square discriminants, 61-62,118-119,123

ternary, 168-169 classically integral, 172 determinant, 169, 171 equivalence, 169 matrix, 169y 2 - xz , 169, 171, 174-175

Quadratic nonresidue, 70, 74, 76-77 Quadratic reciprocity, 63-65, 72-73, 98, 101

proofs, 64, 76, 78-80, 82-83, 84, 91,163-164 supplement, 71, 81. See also Congruences,

quadratic; Legendre symbol, evaluation Quadratic residue, 70, 76-77. See also

Congruences, quadratic; Legendre symbol

Rational approximation, 25,45-47, 50, 105-108, 110, 118

Reduced form, see Reduction, binary forms Reduction, see also Quadratic forms

binary forms, 169negative discriminant, 25, 58-61,181-183

reduced form, 59,168,183-184 positive nonsquare discriminant, 61, 105,

128-132, 141-142and automorphisms, 144 neighbor, 129, 132, 134, 140

modified, 142

and Pell equation, 145 period, 130reduced form, 129, 130-131, 139,142

positive square discriminant, 61-62 zero discriminant, 62

ternary forms, 169-171 Representations of integers, see Diophantine

equations; Prime numbers; Quadratic forms

Riemann, G. F. B., 17 Riemann hypothesis (GRH), 186, 189 Roots of quadratic forms, see Quadratic forms,

binary

Schur, I., 84, 85 Siegel, C. L., 186 Solovay, R., 93-94Squarefree integers, see Integers, squarefree Stark, H. M., 187 Strassen, V., 93-94 Sum of squares:

two, 5, 24, 25, 33, 36-37, 38, 46-47, 49-51, 59, 63, 76, 95,102-103,133-134,168,178

three, 147, 148, 159, 172, 177-178 four, 52-53, 179

Symmetric set, 50

Triangular numbers, 179

Unique factorization, see Factorization, uniqueness

Unique factorization domain, 35, 36

de la Valine Poussin, C. J., 17 Vandermonde matrix, 87, 88 Visible point, 40

Weinberger, P. J., 187,188-189 Wilson’s theorem, 27, 28

generalized, 31Witness, see Composite integer, witness

Zagier, D., 188 Zeta functions, 186

Notation Index

z The ring of integers. C (Z /« ) 84Q The field of rational numbers. Ff 85R T he field o f real numbers. G 88C T he field o f complex numbers. Z [ e 2 n i/" ] 89A x T he group o f units in a g D 90

ring A . °P( a / m ) 92A IX ] The ring o f polynomials in one VA 95

variable with coefficientsA AA 108

in a ring A . 109G L „ (R ) T he group of invertible n X n 109

matrices with entries in R . r x 111G L „(Z ) The group o f n X n matrices Q (v/A) 111

with entries in Z and b\a 2determinant ±1 . GCD 7, 13

S L /f(Z ) The group o f n X n matrices 7 T ( X ) 17with entries in Z and Z / m 25determinant +1 . vm 26

d iag (-) T he block diagonal matrix X 26with specified blocks. s 26

'g T he transpose o f a matrix. Z[/] 34det, tr The determinant and trace o f N 34, 111, 117

a square matrix. A ( F ) 55Im , R e The imaginary and real parts ( « / p ) 70

o f a complex number. m * 73ker The kernel o f a S (n ) 83

homomorphism. o, N 111, 117im The image o f a map. Pa 112, 117deg The degree of a nonzero 113

polynomial. * £ i

212 NOTATION INDEX

( a a , . . . < a „ ) 135

TV 135

»(/)■ «(/) 139r ( « ) , s ( f c ) 143

a ’ tjZ.« 143[a , ft, *J 149

f * g 149

f ' g 149

% 152

fo 152

Errata for Introduction to Number Theory by Daniel E. Flath

Page Line Correction24 −3 qcs1 should read qcss

162 −2 first Cl(∆) should be in the same fontas the (correct) last Cl(∆)

211 20 tg should readtg

212 2 γy should read γy212 6 [a, b,∗ ] should read [a, b, ∗]212 7 f∗g should read f ∗ g212 8 f∼g should read f ∼ g

Updates

There has been a great deal of progress in number theory, both theoreticaland computational, since this book was published 1989, including a proof ofFermat’s Last Theorem. We mention just one result, the resolution by MarkWatkins of the class number 100 problem presented in Appendix A by Serre.Watkins found the complete list of all fundamental negative discriminants withclass number 100 or less, considering all negative discriminants D with |D| ≤2, 383, 747. See Mark Watkins, Mathematics of Computation 73, 2004, pp. 907—938.

1

Growing out of a course designed to teach Gauss’s Disquisitiones Arithmeticaeto honors-level undergraduates, Flath’s Introduction to Number Theory focuses onGauss’s theory of binary quadratic forms. It is suitable for use as a textbook in acourse, or self-study by advanced undergraduates or graduate students who possessa basic familiarity with abstract algebra. The text treats a variety of topics fromelementary number theory including the distribution of primes, sums of squares,continued factions, the Legendre, Jacobi and Kronecker symbols, the class groupand genera. But the focus is on quadratic reciprocity (several proofs are givenincluding one that highlights the p−q symmetry) and binary quadratic forms. Thereader will come away with a good understanding of what Gauss intended in theDisquisitiones and Dirichlet in his Vorlesungen. The text also includes a lovelyappendix by J. P. Serre titled Δ = b2 − 4ac.

The clarity of the author’s vision is matched by the clarity of his exposition.This is a book that reveals the discovery of the quadratic core of algebraic numbertheory. It should be on the desk of every instructor of introductory number theoryas a source of inspiration, motivation, examples, and historical insight.

CHEL/384.H