Higher Arithmetic (Student Mathematical Library)

226 Pages·2007·15.51 MB·English

by  Harold M. Edwards

http://dx.doi.org/10.1090/stml/045
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Higher Arithmeti c
An Algorithmic
Introduction
to Number Theory
STUDENT MATHEMATICAL LIBRARY
Volume 45
Higher Arithmeti c
An Algorithmic
Introduction
to Number Theory
Harold M. Edwards
ilAMS
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island
Editorial Boar d
Gerald B. Folland Bra d G. Osgood (Chair )
Robin Forman Michae l Starbir d
2000 Mathematics Subject Classification. Primar y 11-01 .
For additional information an d updates on this book, visi t
www.ams.org/bookpages/stml-45
Library of Congress Cataloging-in-Publicatio n Dat a
Edwards, Harold M.
Higher arithmetic : an algorithmic introduction to number theory / Harold M.
Edwards.
p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 45)
Includes bibliographical references and index.
ISBN 978-0-8218-4439-7 (alk. paper)
1. Number theory. I . Title.
QA241 .E39 200 8
512.7—dc22 200706057 8
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© 200 8 by the American Mathematical Society. All rights reserved.
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10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 08
Contents
Preface i x
Chapter 1. Number s 1
Chapter 2. Th e Problem AD + B = • 7
Chapter 3. Congruence s 1 1
Chapter 4. Doubl e Congruences and the Euclidean Algorithm 1 7
Chapter 5. Th e Augmented Euclidean Algorithm 2 3
Chapter 6. Simultaneou s Congruences 2 9
Chapter 7. Th e Fundamental Theorem of Arithmetic 3 3
Chapter 8. Exponentiatio n and Orders 3 7
Chapter 9. Euler' s </>-Function 4 3
Chapter 10. Findin g the Order of a mod c 4 5
Chapter 11. Primalit y Testing 5 1
VI Higher Arithmeti c
Chapter 12. The RSA Cipher System 57
Chapter 13. Primitive Roots mod p 61
Chapter 14. Polynomials 67
Chapter 15. Tables of Indices mod p 71
Chapter 16. Brahmagupta's Formula and Hypernumbers 77
Chapter 17. Modules of Hypernumbers 81
Chapter 18. A Canonical Form for Modules of Hypernumbers 87
Chapter 19. Solution of AD + B = • 93
Chapter 20. Proof of the Theorem of Chapter 19 99
Chapter 21. Euler's Remarkable Discovery 113
Chapter 22. Stable Modules 119
Chapter 23. Equivalence of Modules 123
Chapter 24. Signatures of Equivalence Classes 129
Chapter 25. The Main Theorem 135
Chapter 26. Modules That Become Principal When Squared 137
Chapter 27. The Possible Signatures for Certain Values of A 143
Chapter 28. The Law of Quadratic Reciprocity 149
Chapter 29. Proof of the Main Theorem 153
Chapter 30. The Theory of Binary Quadratic Forms 155
Chapter 31. Composition of Binary Quadratic Forms 163
Contents vn
Appendix. Cycle s of Stable Modules 16 9
Answers to Exercises 17 9
Bibliography 20 7
Index 20 9
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Preface
It is widely agreed that Car l Friedrich Gauss's 1801 book Disquisi-
tiones Arithmeticae [G ] was the beginning of modern number theory,
the first wor k on the subject tha t wa s systematic an d comprehen -
sive rather than a collection of special problems and techniques. Th e
name "numbe r theory" by which the subject is known today was in
use at the time—Gauss himself used it (theoria numerorum) i n Arti-
cle 56 of the book—but h e chose to call it "arithmetic " in his title.
He explained i n the first paragraph o f his Preface tha t h e did no t
mean arithmetic in the sense of everyday computations with whole
numbers but a "highe r arithmetic" tha t comprise d "genera l studies
of specific relations among whole numbers."
I too prefer "arithmetic" to "number theory." T o me, number the-
ory sounds passive, theoretical, and disconnected from reality. Higher
arithmetic sounds active, challenging, and related to everyday reality
while aspiring to transcend it.
Although Gauss's explanation of what he means by "higher arith-
metic" in his Preface is unclear, a strong indication of what he had in
mind comes at the end of his Preface when he mentions the material
in his Section 7 on the construction o f regular polygons. (I n mod -
ern terms, Sectio n 7 is the Galois theory of the algebraic equatio n
xn — 1 = 0.) H e admits that this material does not truly belong to
arithmetic but that "it s principles must be drawn from arithmetic. "
IX