This PDF covers the
following topics related to Number Theory : Divisibility, Prime Numbers, The
Linear Diophantine Equation , Congruences, Linear Congruences, The Chinese
Remainder Theorem, Public-Key Cryptography, Pseudoprimes, Polynomial
Congruences with Prime Moduli, Polynomial Congruences with Prime Power
Moduli, The Congruence, General Quadratic Congruences, The Legendre Symbol
and Gauss’ Lemma, Quadratic Reciprocity, Primitive Roots, Arithmetic
Functions, Sums of Squares, Pythagorean Triples, Fermat’s Last Theorem,
Continued Fractions, Simple Continued Fractions, Rational Approximations to
Irrational Numbers, Periodic Continued Fractions, Continued Fraction
Expansion, Pell’s Equation.
This note explains the following topics: Integral ring extensions, Ideals of Dedekind rings, Finiteness
of the class number, Dirichlets unit theorem, Splitting and ramification,
Cyclotomic fields, Valuations and local fields, The theorem of Kronecker
weber.
This note covers the following topics: Primes in
Arithmetic Progressions, Infinite products, Partial summation and Dirichlet
series, Dirichlet characters, L(1, x) and class numbers, The distribution of the
primes, The prime number theorem, The functional equation, The prime number
theorem for Arithmetic Progressions, Siegel’s Theorem, The Polya-Vinogradov
Inequality, Sums of three primes, The Large Sieve, Bombieri’s Theorem.
This note covers the following topics: Divisibility and
Primes, Congruences, Congruences with a Prime-Power Modulus, Euler's Function
and RSA Cryptosystem, Units Modulo an Integer, Quadratic Residues and Quadratic
Forms, Sum of Powers, Fractions and Pell's Equation, Arithmetic Functions, The
Riemann Zeta Function and Dirichlet L-Function.
This is a
textbook about classical elementary number theory and elliptic curves. The first
part discusses elementary topics such as primes, factorization, continued
fractions, and quadratic forms, in the context of cryptography, computation, and
deep open research problems. The second part is about elliptic curves, their
applications to algorithmic problems, and their connections with problems in
number theory.
This note contains the
following subtopics: Classfield theory, homological formulation, harmonic
polynomial multiples of Gaussians, Fourier transform, Fourier inversion on
archimedean and p-adic completions, commutative algebra: integral extensions
and algebraic integers, factorization of some Dedekind zeta functions into
Dirichlet L-functions, meromorphic continuation and functional equation of zeta,
Poisson summation and functional equation of theta, integral representation of
zeta in terms of theta.
Robert Daniel Carmichael (March
1, 1879 – May 2, 1967) was a leading American mathematician.The purpose of this
little book is to give the reader a convenient introduction to the theory of
numbers, one of the most extensive and most elegant disciplines in the whole
body of mathematics. The arrangement of the material is as follows: The five
chapters are devoted to the development of those elements which are essential to
any study of the subject. The sixth and last chapter is intended to give the
reader some indication of the direction of further study with a brief account of
the nature of the material in each of the topics suggested.