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 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.
The
notes contain a useful introduction to important topics that need to be
addressed in a course in number theory. Proofs of basic theorems are presented
in an interesting and comprehensive way that can be read and understood even by
non-majors with the exception in the last three chapters where a background in
analysis, measure theory and abstract algebra is required.
This
note explains the following topics:
Algebraic numbers, Finite continued fractions, Infinite continued fractions,
Periodic continued fractions, Lagrange and Pell, Euler’s totient function,
Quadratic residues and non-residues, Sums of squares and Quadratic forms.