目錄
Preface
1 Natural numbers and integers
1.1 Natural numbers
1.2 Induction
1.3 Integers
1.4 Division with remainder
1.5 Binary notation
1.6 Diophantine equations
1.7 TheDiophantus chord method
1.8 Gaussian integers
1.9 Discussion
2 The Euclidean algorithm
2.1 The gcd by subtraction
2.2 The gcd by division with remainder
2.3 Linear representation of the gcd
2.4 Primes and factorization
2.5 Consequences of unique prime factorization
2.6 Linear Diophantine equations
2.7 The vector Euclidean algorithm
2.8 The map of relatively prime pairs
2.9 Discussion
3 Congruence arithmetic
3.1 Congruence mod n
3.2 Congruence classes and their arithmetic
3.3 Inverses modp
3.4 Fermat's little theorem
3.5 Congruence theorems of Wilson and Lagrange
3.6 Inverses mod k
3.7 Quadratic Diophantine equations
3.8 Primitive roots
3.9 Existence of primitive roots
3.10 Discussion
4 The RSA eryptosystem
4.1 Trapdoor functions
4.2 Ingredients of RSA
4.3 Exponentiation mod n
4.4 RSA encryption and decryption
4.5 Digital signatures
4.6 Other computational issues
4.7 Discussion
5 The Pell equation
5.1 Side and diagonal numbers
5.2 The equation x2-2y2 = 1
5.3 The group of solutions
5.4 The general Pell equation and Z[n]
5.5 The pigeonhole argument
5.6 Quadratic forms
5.7 The map of primitive vectors
5.8 Periodicity in the map ofx2 -ny2
5.9 Discussion
6 The Gaussian integers
6.1 Zand its norm
6.2 Divisibility and primes in Zand Z
6.3 Conjugates
6.4 Division in Z[i]
6.5 Fermat's two square theorem
6.6 Pythagorean triples
6.7 Primes of the form 4n+1
6.8 Discussion
7 Quadratic integers
7.1 The equation y3=x2+2
7.2 The division property in Z[-2]
7.3 The gcd in Z[-2]
7.4 Z[-3] and Z[ζ3]
7.5 Rational solutions of x3+y3=z3+w3
7.6 The prime -3 in Z[ζ3]
7.7 Fermat's last theorem for n=3
7.8 Discussion
8 The four square theorem
8.1 Real matrices and C
8.2 Complex matrices and H
8.3 The quaternion units
8.4 Z[i,j,k]
8.5 The Hurwitz integers
8.6 Conjugates
8.7 A prime divisor property
8.8 Proof of the four square theorem
8.9 Discussion
9 Quadratic reciprocity
9.1 Primes x2+y2, x2+2y2, and x2+3y2
9.2 Statement of quadratic reciprocity
9.3 Euler's criterion
9.4 The value of (2/q)
9.5 The story so far
9.6 The Chinese remainder theorem
9.7 The full Chinese remainder theorem
9.8 Proof of quadratic reciprocity
9.9 Discussion
10 Rings
10.1 The ring axioms
10.2 Rings and fields
10.3 Algebraic integers
10.4 Quadratic fields and their integers
10.5 Norm and units of quadratic fields
10.6 Discussion
11 Ideals
11.1 Ideals and the gcd
11.2 Ideals and divisibility in Z
11.3 Principal ideal domains
11.4 A nonprincipal ideal of Z[-3]
11.5 A nonprincipal ideal of Z[-5]
11.6 Ideals of imaginary quadratic fields as lattices
11.7 Products and prime ideals
11.8 Ideal prime factorization
11.9 Discussion
12 Prime ideals
12.1 Ideals and congruence
12.2 Prime and maximal ideals
12.3 Prime ideals of imaginary quadratic fields
12.4 Conjugate ideals
12.5 Divisibility and containment
12.6 Factorization of ideals
12.7 Ideal classes
12.8 Primes of the form X2+5y2
12.9 Discussion
Bibliography
Index
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