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A Course in Number Theory

Table of Contents

• 1 DIVISIBILITY
  • 1 The Euclidean algorithm and unique factorization
  • 2 Prime numbers
  • 3 Problems 2
• 2 MULTIPLICATIVE FUNCTIONS
  • 1 The Möbius and Euler functions
  • 2 Average order
  • 3 Problems 3
• 3 CONGRUENCE THEORY
  • 1 Definitions and linear congruences
  • 2 Nonlinear congruences and the theorems of Euler, Lagrange, and Chevalley
  • 3 Local versus global considerations
  • 4 Computation modulo n
  • 5 Problems 3
• 4 QUADRATIC RESIDUES
  • 1 The Legendre symbol
  • 2 Quadratic reciprocity
  • 3 Some further topics
  • Problems 4
• 5 ALGEBRAIC TOPICS
  • 1 Algebraic numbers and integers
  • 2 Primitive roots
  • 3 Characters
  • 4 Problems 5
• 6 SUMS OF SQUARES AND GAUSS SUMS
  • 1 Sums of squares
  • 2 Gauss and Jacobi sums
  • 3 The sign of the quadratic Gauss sum
  • 4 Problems 6
• 7 CONTINUED FRACTIONS
  • 1 Basic properties
  • 2 Best approximation
  • 3 Pell’s equation
  • 4 A set of real numbers modulo 1
  • 5 Problems 7
• 8 TRANSCENDENTAL NUMBERS
  • 1 Liouville’s theorem and applications
  • 2 The Hermite and Lindemann theorems
  • 3 The Gelfond-Schneider theorem
  • 4 Problems 8
• 9 QUADRATIC FORMS
  • 1 Equivalence of forms
  • 2 Sums of three squares
  • 3 Representation by binary forms
  • 4 Algorithms for reduced forms
  • 5 Problems 9
• 10 GENERA AND THE CLASS GROUP
  • 1 The genus of a form
  • 2 Composition and the class group
  • 3 A formula for the class number
  • 4 Problems 10
• 11 PARTITIONS
  • 1 Elementary properties
  • 2 Jacobi’s identity
  • 3 Estimates for p(n)
  • 4 Problems 11
• 12 THE PRIME NUMBERS
  • 1 The results of Chebyshev and Bertrand
  • 2 Series involving primes
  • 3 Riemann zeta function
  • 4 Problems 12
• 13 TWO MAJOR THEOREMS ON THE PRIMES
  • 1 Dirichlet’s theorem
  • 2 PNT: preliminaries and Selberg’s theorem
  • 3 PNT: the main proof
  • 4 Problems 13
• 14 DIOPHANTINE EQUATIONS
  • 1 Legendre’s theorem
  • 2 Fermat’s last theorem
  • 3 Skolem’s method
  • 4 Mordell’s equation
  • 5 Problems 14
• 15 ELLIPTIC CURVES: BASIC THEORY
  • 1 Geometric preliminaries
  • 2 Rational points on elliptic curves
  • 3 Mordell-Weil theorem
  • 4 Problems 15
• 16 ELLIPTIC CURVES: FURTHER RESULTS AND APPLICATIONS
  • 1 Weierstrass equations
  • 2 Nagell-Lutz theorem
  • 3 Curves defined over finite fields
  • 4 Lenstra’s factorization method
  • 5 L-functions for curves
  • 6 Problems 16
• ANSWERS AND HINTS TO PROBLEMS
  • Problems 1
  • Problems 2
  • Problems 3
  • Problems 4
  • Problems 5
  • Problems 6
  • Problems 7
  • Problems 8
  • Problems 9
  • Problems 10
  • Problems 11
  • Problems 12
  • Problems 13
  • Problems 14
  • Problems 15
  • Problems 16
• TABLES
• BIBLIOGRAPHY
• INDEX OF NOTATION
• GENERAL INDEX

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