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El. knyga: Elementary Number Theory

(Skidmore College, New York, USA), (Penn State University, University Park, Pennsylvania, USA)
  • Formatas: 254 pages
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 08-Sep-2021
  • Leidėjas: Chapman & Hall/CRC
  • Kalba: eng
  • ISBN-13: 9781000427110
Kitos knygos pagal šią temą:
Elementary Number TheoryDidesnis paveikslėlis
  • Formatas: 254 pages
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 08-Sep-2021
  • Leidėjas: Chapman & Hall/CRC
  • Kalba: eng
  • ISBN-13: 9781000427110
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"This text is intended to be used as an undergraduate introduction to the theory of numbers. The authors have been immersed in this area of mathematics for many years and hope that this text will inspire students (and instructors) to study, understand, and come to love this truly beautiful subject"--

This text is intended to be used as an undergraduate introduction to the theory of numbers. The authors have been immersed in this area of mathematics for many years and hope that this text will inspire students (and instructors) to study, understand, and come to love this truly beautiful subject.



Elementary Number Theory, Gove Effinger, Gary L. Mullen

This text is intended to be used as an undergraduate introduction to the theory of numbers. The authors have been immersed in this area of mathematics for many years and hope that this text will inspire students (and instructors) to study, understand, and come to love this truly beautiful subject.

Each chapter, after an introduction, develops a new topic clearly broken out in sections which include theoretical material together with numerous examples, each worked out in considerable detail. At the end of each chapter, after a summary of the topic, there are a number of solved problems, also worked out in detail, followed by a set of supplementary problems. These latter problems give students a chance to test their own understanding of the material; solutions to some but not all of them complete the chapter.

The first eight chapters discuss some standard material in elementary number theory. The remaining chapters discuss topics which might be considered a bit more advanced. The text closes with a chapter on Open Problems in Number Theory. Students (and of course instructors) are strongly encouraged to study this chapter carefully and fully realize that not all mathematical issues and problems have been resolved! There is still much to be learned and many questions to be answered in mathematics in general and in number theory in particular.

Preface ix
1 Divisibility in the Integers Z
1(14)
1.1 Introduction
1(1)
1.2 Divisibility
1(1)
1.3 The Division Algorithm
2(1)
1.4 Greatest Common Divisors
3(1)
1.5 The Euclidean Algorithm
3(3)
1.6 Summary
6(1)
1.7 Solved Problems
7(3)
1.8 Supplementary Problems
10(5)
2 Prime Numbers and Factorization
15(14)
2.1 Introduction
15(1)
2.2 Identifying Primes
16(1)
2.3 [ Listing Primes: The Sieve of Eratosthenes
16(2)
2.4 Unique Factorization of Integers into Primes
18(3)
2.5 The Difficulty of Factorization
21(1)
2.6 Using Factorization to Compute a GCD
21(1)
2.7 Summary
22(1)
2.8 Solved Problems
23(3)
2.9 Supplementary Problems
26(3)
3 Congruences and the Sets Zn
29(16)
3.1 Introduction
29(1)
3.2 Definition and Examples of Congruences
29(1)
3.3 The Finite Sets Zn
30(1)
3.4 Addition and Multiplication Tables for Zn
31(1)
3.5 Properties of Congruences
32(2)
3.6 Doing Division in Zn
34(3)
3.7 Summary
37(1)
3.8 Solved Problems
38(3)
3.9 Supplementary Problems
41(4)
4 Solving Congruences
45(16)
4.1 Introduction
45(1)
4.2 Solving a Single Linear Congruence
45(4)
4.3 Solving Systems of Two or More Congruences
49(4)
4.4 Summary
53(1)
4.5 Solved Problems
54(4)
4.6 Supplemental Problems
58(3)
5 The Theorems of Fermat and Euler
61(18)
5.1 Introduction
61(1)
5.2 Fermat's Theorem for Prime Moduli
61(3)
5.3 Euler's Function and Euler's Theorem
64(5)
5.4 Fast Exponentiation
69(2)
5.5 Summary
71(1)
5.6 Solved Problems
71(4)
5.7 Supplementary Problems
75(4)
6 Applications to Modern Cryptography
79(20)
6.1 Introduction
79(1)
6.2 The Basics of Encryption
80(1)
6.3 Primitive Roots in Zp
81(1)
6.4 Diffie-Hellman Key Exchange
82(3)
6.5 Public Key Cryptography and the RSA System
85(4)
6.6 Security versus Authenticity
89(2)
6.7 Summary
91(1)
6.8 Solved Problems
92(3)
6.9 Supplemental Problems
95(4)
7 Quadratic Residues and Quadratic Reciprocity
99(20)
7.1 Introduction
99(1)
7.2 Quadratic Residues and the Legendre Symbol
99(3)
7.3 Computing the Legendre Symbol
102(3)
7.4 Quadratic Reciprocity
105(3)
7.5 Composite Moduli and the Jacobi Symbol
108(2)
7.6 Summary
110(1)
7.7 Solved Problems
111(5)
7.8 Supplementary Problems
116(3)
8 Some Fundamental Number Theory Functions
119(20)
8.1 Introduction
119(1)
8.2 The Greatest Integer Function
119(4)
8.3 The Functions τ(η), σ(η), σκ(η)
123(4)
8.4 The Mobius Inversion Formula
127(3)
8.5 Summary
130(1)
8.6 Solved Problems
131(4)
8.7 Supplementary Problems
135(4)
9 Diophantine Equations
139(26)
9.1 Introduction
139(1)
9.2 The Linear Equation ax + by = c
139(3)
9.3 The Equation x2 + y2 = z2
142(3)
9.4 The Equation x4 + y4 = z4
145(3)
9.5 The Equation xn + yn = zn, n < 2
148(1)
9.6 Sums of Four Squares
149(7)
9.7 Waring's Problem
156(1)
9.8 Summary
157(1)
9.9 Solved Problems
158(3)
9.10 Supplementary Problems
161(4)
10 Finite Fields
165(32)
10.1 Introduction
165(1)
10.2 The Finite Fields Fpn
166(1)
10.3 The Order of a Finite Field
167(3)
10.4 Constructing Finite Fields
170(4)
10.5 The Multiplicative Structure of Fq
174(3)
10.6 The Subfields of Fpn
177(2)
10.7 Counting Irreducible Polynomials over Fpn
179(4)
10.8 Lagrange Interpolation Formula
183(2)
10.9 An Application to Latin and Sudoku Squares
185(4)
10.10 Summary
189(1)
10.11 Solved Problems
190(4)
10.12 Supplementary Problems
194(3)
11 Some Open Problems in Number Theory
197(20)
11.1 Introduction
197(1)
11.2 Open Problems
198(13)
11.3 Summary
211(1)
11.4 Problems
211(6)
A Mathematical Induction 217(8)
B Sets of Numbers Beyond the Integers 225(12)
Bibliography 237(2)
Index 239
Gove Effinger received his Ph.D. in Mathematics from the University of Massachusetts (Amherst) in 1981 and subsequently taught at Bates College for 5 years and then Skidmore College for 29 years. He is the author of three books: An Elementary Transition to Abstract Mathematics (with Gary L. Mullen, CRC Press), Additive Number Theory of Polynomials over a Finite Field (with David R. Hayes), and Common-Sense BASIC: Structured Programming with Microsoft Quick BASIC (with Alice M. Dean), as well as numerous research papers. His research focus has primarily been concerned with the similarities of polynomials over a finite fields and ordinary integers.









Gary L. Mullen is Professor of Mathematics at the Pennsylvania State University, University Park, PA. He has taught both undergraduate and graduate courses there for over 40 years. In addition, he has written more than 150 research papers and co-authored seven books, including both graduate as well as undergraduate textbooks. He also served as department head for seven years and has served as an editor on numerous editorial boards, including having served as Editor-in-Chief of the journal Finite Fields and Their Applications since its founding in 1995.
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