Introduction
Ⅰ Factorization and the Primes
1.The laws of arithmetic
2.Proof by induction
3.Prime numbers
4.The fundamental theorem of arithmetic
5.Consequences of the fundamental theorem
6.Euclid's algorithm
7.Another proof of the fundamental theorem
8.A property of the H.C.F
9.Factorizing a number
10.The series of primes
Ⅱ Congruences
1.The congruence notation
2.Linear congruences
3.Fermat's theorem
4.Euler's function φ（m）
5.Wilson's theorem
6.Algebraic congruences
7.Congruences to a prime modulus
8.Congruences in several unknowns
9.Congruences covering all numbers
Ⅲ Quadratic Residues
1.Primitive roots
2.Indices
3.Quadratic residues
4.Gauss's lemma
5.The law of reciprocity
6.The distribution of the quadratic residues
Ⅳ Continued Fractions
1.Introduction
2.The general continued fraction
3.Euler's rule
4.The convergents to a continued traction
5.The equation ax-by=1
6.Infinite continued fractions
7.Diophantine approximation
8.Quadratic irrationals
9.Purely periodic continued fractions
10.Lagrange's theorem
11.Pell's equation
12.A geometricalinterpretation of continued fractions
Ⅴ Sums of Squares
1.Numbers representable by two squares
2.Primes of the form 4k+1
3.Constructions for x and y
4.Representation by four squares
5.Representation by three squares
Ⅵ Quadratic Forms
1.Introduction
2.Equivalent forms
3.The discriminant
4.The representation of a number by a form
5.Three examples
6.The reduction of positive definite forms
7.The reduced forms
8.The number of representations
9.The class-number
Ⅶ Some Diophantine Equations
1.Introduction
2.The equation x2+y2=z2
3.The equation ax2+by2=z2
4.Elliptic equations and curves
5.Elliptic equations modulo primes
6.Fermat's Last Theorem
7.The equation x3+y3=z3+w3
8.Further developments
Ⅷ Computers and Number Theory
1.Introduction
2.Testing for primality
3.‘Random’ number generators
4.Pollard's factoring methods
5.Factoring and primality via elliptic curves
6.Factoring large numbers
7.The Diffie-Hellman cryptographic method
8.The RSA cryptographic method
9.Primality testing revisited
Exercises
Hints
Answers
Bibliography
Index
