An Introduction to Number Theory

by Everest, Graham

• ISBN13: 9781852339173
• ISBN10: 1852339179
• Format: Hardcover
• Copyright: 2005-05-30
• Publisher: Springer Nature

Summary

An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory. A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography. Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory.

Table of Contents

Introduction	1	(206)
1 A Brief History of Prime	7	(36)
1.1 Euclid and Primes	7	(4)
1.2 Summing Over the Primes	11	(5)
1.3 Listing the Primes	16	(13)
1.4 Fermat Numbers	29	(2)
1.5 Primality Testing	31	(4)
1.6 Proving the Fundamental Theorem of Arithmetic	35	(4)
1.7 Euclid's Theorem Revisited	39	(4)
2 Diophantine Equations	43	(16)
2.1 Pythagoras	43	(2)
2.2 The Fundamental Theorem of Arithmetic in Other Contexts	45	(3)
2.3 Sums of Squares	48	(4)
2.4 Siegel's Theorem	52	(4)
2.5 Fermat, Catalan, and Euler	56	(3)
3 Quadratic Diophantine Equations	59	(24)
3.1 Quadratic Congruences	59	(6)
3.2 Euler's Criterion	65	(2)
3.3 The Quadratic Reciprocity Law	67	(6)
3.4 Quadratic Rings	73	(2)
3.5 Units in Z[square root of d], d > 0	75	(3)
3.6 Quadratic Forms	78	(5)
4 Recovering the Fundamental Theorem of Arithmetic	83	(10)
4.1 Crisis	83	(1)
4.2 An Ideal Solution	84	(1)
4.3 Fundamental Theorem of Arithmetic for Ideals	85	(4)
4.4 The Ideal Class Group	89	(4)
5 Elliptic Curves	93	(28)
5.1 Rational Points	93	(5)
5.2 The Congruent Number Problem	98	(7)
5.3 Explicit Formulas	105	(5)
5.4 Points of Order Eleven	110	(2)
5.5 Prime Values of Elliptic Divisibility Sequences	112	(5)
5.6 Ramanujan Numbers and the Taxicab Problem	117	(4)
6 Elliptic Functions	121	(12)
6.1 Elliptic Functions	121	(5)
6.2 Parametrizing an Elliptic Curve	126	(2)
6.3 Complex Torsion	128	(1)
6.4 Partial Proof of Theorem 6.5	129	(4)
7 Heights	133	(24)
7.1 Heights on Elliptic Curves	133	(5)
7.2 Mordell's Theorem	138	(4)
7.3 The Weak Mordell Theorem: Congruent Number Curve	142	(4)
7.4 The Parallelogram Law and the Canonical Height	146	(4)
7.5 Mahler Measure and the Naive Parallelogram Law	150	(7)
8 The Riemann Zeta Function	157	(26)
8.1 Euler's Summation Formula	158	(3)
8.2 Multiplicative Arithmetic Functions	161	(3)
8.3 Dirichlet Convolution	164	(5)
8.4 Euler Products	169	(2)
8.5 Uniform Convergence	171	(2)
8.6 The Zeta Function Is Analytic	173	(2)
8.7 Analytic Continuation of the Zeta Function	175	(8)
9 The Functional Equation of the Riemann Zeta Function	183	(24)
9.1 The Gamma Function	183	(2)
9.2 The Functional Equation	185	(2)
9.3 Fourier Analysis on Schwartz Spaces	187	(2)
9.4 Fourier Analysis of Periodic Functions	189	(5)
9.5 The Theta Function	194	(3)
9.6 The Gamma Function Revisited	197	(10)
10 Primes in an Arithmetic Progression	207	(18)
10.1 A New Method of Proof	208	(3)
10.2 Congruences Modulo 3	211	(2)
10.3 Characters of Finite Abelian Groups	213	(4)
10.4 Dirichlet Characters and L-Functions	217	(2)
10.5 Analytic Continuation and Abel's Summation Formula	219	(4)
10.6 Abel's Limit Theorem	223	(2)
11 Converging Streams	225	(20)
11.1 The Class Number Formula	225	(4)
11.2 The Dedekind Zeta Function	229	(4)
11.3 Proof of the Class Number Formula	233	(2)
11.4 The Sign of the Gauss Sum	235	(3)
11.5 The Conjectures of Birch and Swinnerton-Dyer	238	(7)
12 Computational Number Theory	245	(34)
12.1 Complexity of Arithmetic Computations	245	(6)
12.2 Public-key Cryptography	251	(2)
12.3 Primality Testing: Euclidean Algorithm	253	(5)
12.4 Primality Testing: Pseudoprimes	258	(2)
12.5 Carmichael Numbers	260	(2)
12.6 Probabilistic Primality Testing	262	(4)
12.7 The Agrawal-Kayal-Saxena Algorithm	266	(3)
12.8 Factorizing	269	(7)
12.9 Complexity of Arithmetic in Finite Fields	276	(3)
References	279	(8)
Index	287