Shanks Daniel - Solved and unsolved problems in Number Theory.pdf - Number Theory - jacpr

CONTENTS

PAGE

PREFACE

.......................................

Chapter I

FROM PERFECT KGXIBERS TO THE QUADRATIC

RECIPROCITY LAW

SECTION

1 . Perfect Xumbcrs ..........................................

.............................

SECOND EDITION

................ ...............

4 . Euclid’s Algorithm ....... ...............................

5 . Cataldi and Others ...... ...............................

6 . The Prime Kumber Theorem ..............................

7 . Two Useful Theorems ......................................

8 Fermat.and 0t.hcrs

........................................

9 . Euler’s Generalization Promd ...............................

10 Perfect Kunibers, I1

.......................................

11 . Euler and dial.. ...........................

Library of Congress Cataloging in Publication Data

Shanks. Daniel .

Solved and unsolved problems in number theory .

Bibliography: p .

Includes index .

.............

12 . Many Conjectures and their Interrelations ....................

13 . Splitting tshePrimes into Equinumerous Classes ...............

14 . Euler’s Criterion Formulated ...... .......................

15 . Euler’s Criterion Proved ....................................

16 . Wilson’s Theorem .........................................

1 .

Numbers. Theory of .

I .

Title .

17 . Gauss’s Criterion

...................................

18 . The Original Lcgendre Symbol ..

[QA241.S44

19781

5E.7

77-13010

.....................

19 . The Reciprocity Law ..........

ISBN 0-8284-0297-3

20 . The Prime Divisors of n2 + a ...

.....................

Printed on ‘long-life’ acid-free paper

Chapter I1

THE CSDERLTISG STRUCTURE

21 . The Itesidue Classes as an Invention

22 . The Residue Classes 3s a Tool ..............................

23 . The Residue Classcs as n Group .............................

24 . Quadratic Residues

.........................

Printed in the United States of America

.......................................

2 . Euclid ............

3 . Euler’s Converse Pr

Solved and Unsolved Problems in Number Theory

Contents

vii

SECTION

PAGE

6.2

SECTION PAGE

61 . Continued Fractions for fi ............................... 180

62 . From Archimedes to Lucas ................................. 188

63 . The Lucas Criterion ....................................... 193

64 . A Probability Argument .................................... 197

65 . Fibonacci Xumbers and the Original Lucas Test .............. 198

25 . Is the Quadratic Recipro&y Law a Ilerp Thcoreni?

26 . Congruent. id Equations with a Prime Modulus

27 . Euler’s 4 E’unct.ion .........................................

...........

................

28 . Primitive Roots with a Prime i\Iodulus .

29 . mp as a Cyclic Group ................

...........

........... 73

30 . The Circular Parity Switch ...........

...........

31 . Primitive Roots and Fermat Xumhcrs.,

........... 78

32 . Artin’s Conjectures .................. ........... 80

33 . Questions Concerning Cycle Graphs .... ........... 83

34 . Answers Concerning Cycle Graphs ..... ........... 92

35 . Factor Generators of 3% ...................................

36 . Primes in Some Arithmetic Progressions and a General Divisi-

bility Theorem .......... ............................

Appendix to Chapters 1-111

SUPPLEMENTARY

COMMENTS.

THEOREMS.

AND EXERCISES

......... 201

Chapter IV

PROGRESS

104

37 . Scalar and Vect.or Indices ...................................

109

SECTION

66 . Chapter I Fifteen Years Later ....................

......................

113

217

67 . Artin’s Conjectures, I1 ......................... 222

68 . Cycle Graphs and Related Topics ................... 225

39 . The Converse of Fermat’s Theore

................

40 . Sufficient Coiiditiorls for Primality ...........................

118

69 . Pseudoprimes and Primality ......................

226

70 . Fermat’s Last “Theorem, ” I1 .....................

231

Chapter 111

PYTHAGOREAKISM AKD ITS MAXY COKSEQUESCES

41 . The Pythagoreans ................... ............ 121

42 . The Pythagorean Theorein ........... ................ 123

43 . The 42 and the Crisis ..........................

44 . The Effect upon Geometry .................................

71 . Binary Quadratic Forms with Negative Discriminants ...... 233

72 . Binary Quadratic Forms with Positive Discriminants ....... 235

73 . Lucas and Pythagoras .........................

237

74 . The Progress Report Concluded ................... 238

127

45 . The Case for Pythagoreanism .........................

46 . Three Greek Problems ..... ..................

47 . Three Theorems of Fermat .................................

Appendix

48 . Fermat’s Last’ “Theorem” ....... .......................

142

STATEMENT

ON FUNDAMENTALS

.......................

239

144

TABLE OF DEFINITIONS

............................

241

49 . The Easy Case and Infinite Desce

.......................

147

50 . Gaussian Integers and Two Applications .

........... 149

REFERENCES

...................................

243

......................................

255

51 . Algebraic Integers and Kummer’s Theore

........... 1.51

52 . The Restricted Case, S

53 . Euler’s “Conjecture” . .

Gcrmain, and Wieferich ...

................................

157

54 . Sum of Two Squares .......................

...... 159

55 . A Generalization and Geometric Xumber Theory .............. 161

........ 165

56 . A Generalization and Binary Quadratic Forms .....

57 . Some Applicat.ions ..............................

58 . The Significance of Fermat’s Equation ............

59 . The Main Theorem ........... ..........................

60 . An Algorithm .................................

..... 178

174

38 . The Ot. her Residue Classes .......

INDEX

PREFACE TO THE SECOND EDITION

The Preface to the First Edition (1962) states that this is “a rather

tightly organized presentation of elementary number theory” and that

“number theory is very much a live subject.” These two facts are in

conflict fifteen years later. Considerable updating is desirable at many

places in the 1962 Gxt, but the needed insertions would call for drastic

surgery. This could easily damage the flow of ideas and the author was

reluctant to do that. Instead, the original text has been left as is, except

for typographical corrections, and a brief new chapter entitled “Pro-

gress” has been added. A new reader will read the book at two

levels-as it was in 1962, and as things are today.

Of course, not all advances in number theory are discussed, only those

pertinent to the earlier text. Even then, the reader will be impressed

with the changes that have occurred and will come to believe-if he did

not already know it-that number theory is very much a live subject.

The new chapter is rather different in style, since few topics are

developed at much length. Frequently, it is extremely hrief and merely

gives references. The intent is not only to discuss the most important

changes in sufficient detail but also to be a useful guide to many other

topics. A propos this intended utility, one special feature: Developments

in the algorithmic and computational aspects of the subject have been

especially active. It happens that the author was an editor of Muthe-

matics of Cmptation throughout this period, and so he was particu-

larly close to most of these developments. Many good students and

professionals hardly know this material at all. The author feels an

obligation to make it better known, and therefore there is frequent

emphasis on these aspects of the subject.

To compensate for the extreme brevity in some topics, numerous

references have been included to the author’s own reviews on these

topics. They are intended especially for any reader who feels that he

must have a second helping. Many new references are listed, but the

following economy has been adopted: if a paper has a good bibliogra-

phy, the author has usually refrained from citing the references con-

tained in that bibliography.

The author is grateful to friends who read some or all of the new

chapter. Especially useful comments have come from Paul Bateman,

Samuel Wagstaff, John Brillhart, and Lawrence Washington.

DANIELSHANKS

December 1977

PREFACE TO THE FIRST EDITION

the book is, in fact, a rather tightly organized presentation of elementary

number theory, while the title may suggest a loosely organized collection

of problems. h-onetheless the nature of the exposition and the choice of

topics to be included or. omitted are such as to make the title appropriate.

Since a preface is the proper place for such discussion we wish to clarify

this matter here.

Much of elementary number theory arose out of the investigation of

three problems ; that of perfect numbers, that of periodic decimals, and

that of Pythagorean numbers. We have accordingly organized the book

into three long chapters. The result of such an organization is that motiva-

tion is stressed to a rather unusual degree. Theorems arise in response to

previously posed problems, and their proof is sometimes delayed until

an appropriate analysis can be developed. These theorems, then, or most

of them, are “solved problems.” Some other topics, which are often taken

up in elementary texts-and often dropped soon after-do not fit directly

into these main lines of development, and are postponed until Volume 11.

i ~

Since number theory is so extensive, some choice of topics is essential, and

while a common criterion used is the personal preferences or accomplish-

ments of an author, there is available this other procedure of following,

rather closely, a few main themes and postponing other topics until they

become necessary.

Historical discussion is, of course, natural in such a presentation. How-

ever, our primary interest is in the theorems, and their logical interrela-

tions, and not in the history per se. The aspect of the historical approach

which mainly concerns us is the determination of the problems which sug-

gested the theorems, and the study of which provided the concepts and the

techniques which were later used in their proof. In most number theory

books residue classes are introduced prior to Fermat’s Theorem and the

Reciprocity Law. But this is not at all the correct historical order. We have

here restored these topics to their historical order, and it seems to us that

this restoration presents matters in a more natural light.

The “unsolved problems” are the conjectures and the open questions-

we distinguish these two categories-and these problems are treated more

fully than is usually the caw. The conjectures, like the theorems, are in-

troduced at the point at which they arise naturally, are numbered and

stated formally. Their significance, their interrelations, and the heuristic

It may be thought that the title of this book is not well chosen since

Shanks Daniel - Solved and unsolved problems in Number Theory.pdf

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