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Number Theory Elementary Number Theory by Gareth A. Jones

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  1. Jan 19, 2013 #1

    Table of Contents:
    Code (Text):

    [LIST]
    [*] Notes to the Reader
    [*] Divisibility
    [LIST]
    [*] Divisors
    [*] Bezout's identity
    [*] Least common multiples
    [*] Linear Diophantine equations
    [*] Supplementary exercises
    [/LIST]
    [*] Prime Numbers
    [LIST]
    [*] Prime numbers and prime-power factorisations
    [*] Distribution of primes
    [*] Fermat and Mersenne primes
    [*] Primality-testing and factorisation
    [*] Supplementary exercises
    [/LIST]
    [*] Congruences
    [LIST]
    [*] Modular arithmetic
    [*] Linear congruences
    [*] Simultaneous linear congruences
    [*] Simultaneous non-linear congruences
    [*] An extension of the Chinese Remainder Theorem
    [*] Supplementary exercises
    [/LIST]
    [*] Congruences with a Prime-power Modulus
    [LIST]
    [*] The arithmetic of [itex]\mathbb{Z}_p[/itex]
    [*] Pseudoprimes and Carmichael numbers
    [*] Solving congruences mod [itex](p^e)[/itex]
    [*] Supplementary exercises
    [/LIST]
    [*] Euler's Function
    [LIST]
    [*] Units
    [*] Euler's function
    [*] Applications of Euler's function
    [*] Supplementary exercises
    [/LIST]
    [*] The Group of Units
    [LIST]
    [*] The group [itex]U_n[/itex]
    [*] Primitive roots
    [*] The group [itex]U_{p^e}[/itex], where [itex]p[/itex] is an odd prime
    [*] The group [itex]U_{2^e}[/itex]
    [*] The existence of primitive roots
    [*] Applications of primitive roots
    [*] The algebraic structure of [itex]U_n[/itex]
    [*] The universal exponent
    [*] Supplementary exercises
    [/LIST]
    [*] Quadratic Residues
    [LIST]
    [*] Quadratic congruences
    [*] The group of quadratic residues
    [*] The Legendre symbol
    [*] Quadratic reciprocity
    [*] Quadratic residues for prime-power moduli
    [*] Quadratic residues for arbitrary moduli
    [*] Supplementary exercises
    [/LIST]
    [*] Arithmetic Functions
    [LIST]
    [*] Definition and examples
    [*] Perfect numbers
    [*] The Mobius Inversion Formula
    [*] An application of the Mobius Inversion Formula
    [*] Properties of the Mobius function
    [*] The Dirichlet product
    [*] Supplementary exercises
    [/LIST]
    [*] The Riemann Zeta Function
    [LIST]
    [*] Historical background
    [*] Convergence
    [*] Applications to prime numbers
    [*] Random integers
    [*] Evaluating [itex]\zeta(2)[/itex]
    [*] Evaluating [itex]\zeta(2k)[/itex]
    [*] Dirichlet series
    [*] Euler products
    [*] Complex variables
    [*] Supplementary exercises
    [/LIST]
    [*] Sums of Squares
    [LIST]
    [*] Sums of two squares
    [*] The Gaussian integers
    [*] Sums of three squares
    [*] Sums of four squares
    [*] Digression on quaternions
    [*] Minkowski's Theorem
    [*] Supplementary exercises
    [/LIST]
    [*] Fermat's Last Theorem
    [LIST]
    [*] The problem
    [*] Pythagoras's Theorem
    [*] Pythagorean triples
    [*] Isosceles triangles and irrationality
    [*] The classification of Pythagorean triples
    [*] Fermat
    [*] The case [itex]n = 4[/itex]
    [*] Odd prime exponents
    [*] Lame and Kummer
    [*] Modern developments
    [*] Further reading
    [/LIST]
    [*] Appendix: Induction and Well-ordering
    [*] Appendix: Groups, Rings and Fields
    [*] Appendix: Convergence
    [*] Appendix: Table of Primes [itex]p < 1000[/itex]
    [*] Solutions to Exercises
    [*] Bibliography
    [*] Index of symbols
    [*] Index of names
    [*] Index
    [/LIST]
     
     
    Last edited: May 6, 2017
  2. jcsd
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