Elementary Number Theory /

Our intention in writing this book is to give an elementary introduction to number theory which does not demand a great deal of mathematical back­ ground or maturity from the reader, and which can be read and understood with no extra assistance. Our first three chapters are based almost entirely on...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Jones, Gareth A. (Autor), Jones, J. Mary (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: London : Springer London : Imprint : Springer, 1998.
Colección:Springer undergraduate mathematics series,
Temas:
Mathematics.
Number theory.
Mathématiques.
Théorie des nombres.
mathematics.
applied mathematics.
Matemáticas
Números , Teoría de
Mathematics
Number theory
Zahlentheorie
Nombres, Théorie des.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1. Divisibility
  • 1.1 Divisors
  • 1.2 Bezout's identity
  • 1.3 Least common multiples
  • 1.4 Linear Diophantine equations
  • 1.5 Supplementary exercises
  • 2. Prime Numbers
  • 2.1 Prime numbers and prime-power factorisations
  • 2.2 Distribution of primes
  • 2.3 Fermat and Mersenne primes
  • 2.4 Primality-testing and factorisation
  • 2.5 Supplementary exercises
  • 3. Congruences
  • 3.1 Modular arithmetic
  • 3.2 Linear congruences
  • 3.3 Simultaneous linear congruences
  • 3.4 Simultaneous non-linear congruences
  • 3.5 An extension of the Chinese Remainder Theorem
  • 3.6 Supplementary exercises
  • 4. Congruences with a Prime-power Modulus
  • 4.1 The arithmetic of?p
  • 4.2 Pseudoprimes and Carmichael numbers
  • 4.3 Solving congruences mod (pe)
  • 4.4 Supplementary exercises
  • 5. Euler's Function
  • 5.1 Units
  • 5.2 Euler's function
  • 5.3 Applications of Euler's function
  • 5.4 Supplementary exercises
  • 6. The Group of Units
  • 6.1 The group Un
  • 6.2 Primitive roots
  • 6.3 The group Une, where p is an odd prime
  • 6.4 The group U2e
  • 6.5 The existence of primitive roots
  • 6.6 Applications of primitive roots
  • 6.7 The algebraic structure of Un
  • 6.8 The universal exponent
  • 6.9 Supplementary exercises
  • 7. Quadratic Residues
  • 7.1 Quadratic congruences
  • 7.2 The group of quadratic residues
  • 7.3 The Legendre symbol
  • 7.4 Quadratic reciprocity
  • 7.5 Quadratic residues for prime-power moduli
  • 7.6 Quadratic residues for arbitrary moduli
  • 7.7 Supplementary exercises
  • 8. Arithmetic Functions
  • 8.1 Definition and examples
  • 8.2 Perfect numbers
  • 8.3 The Mobius Inversion Formula
  • 8.4 An application of the Mobius Inversion Formula
  • 8.5 Properties of the Mobius function
  • 8.6 The Dirichlet product
  • 8.7 Supplementary exercises
  • 9. The Riemann Zeta Function
  • 9.1 Historical background
  • 9.2 Convergence
  • 9.3 Applications to prime numbers
  • 9.4 Random integers
  • 9.5 Evaluating?(2)
  • 9.6 Evaluating?(2k)
  • 9.7 Dirichlet series
  • 9.8 Euler products
  • 9.9 Complex variables
  • 9.10 Supplementary exercises
  • 10. Sums of Squares
  • 10.1 Sums of two squares
  • 10.2 The Gaussian integers
  • 10.3 Sums of three squares
  • 10.4 Sums of four squares
  • 10.5 Digression on quaternions
  • 10.6 Minkowski's Theorem
  • 10.7 Supplementary exercises
  • 11. Fermat's Last Theorem
  • 11.1 The problem
  • 11.2 Pythagoras's Theorem
  • 11.3 Pythagorean triples
  • 11.4 Isosceles triangles and irrationality
  • 11.5 The classification of Pythagorean triples
  • 11.6 Fermat
  • 11.7 The case n = 4
  • 11.8 Odd prime exponents
  • 11.9 Lame and Kummer
  • 11.10 Modern developments
  • 11.11 Further reading
  • Solutions to Exercises
  • Index of symbols
  • Index of names.

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