Number Theory

• MAT21300T, Summer

Introduction

1. Number theory
2. Mathematicians
3. Diophantine Equations

Natural Number and Integer

1. Successor Function, Peano Axioms
2. Principle of Mathematical Induction (PMI), Well Ordering Principle (WOP)
3. Divisibility, Division with Remainder, Greatest Common Divisor (GCD), Euclidean Algorithm
4. Co-Prime (or Relatively Prime), Prime Number, Fundamental Theorem of Arithmetic,
5. Infinitely Many Primes, Goldbach Conjecture, Twin Prime Conjecture, Mersenne Prime Conjecture

Modular Arithmetic

1. Congruence, Addition, Subtraction, Multiplication, Division (ModInverse)
2. Complete Residue System, Reduced Residue System, Euler's Totient Function
3. Euler's Theorem, Fermat's Little Theorem
4. Inverse of Elements, Division, Wilson's Theorem
5. Congruence Equation, Linear Congruence Equation
6. Linear Congruences, Chinese Remainder Theorem (CRT)

Prime Power and Primitive Root

1. Primality Testing, Carmichael Number, Factorization
2. Hensel's Lemma
3. Order of Element, Primitive Powers, Primitive Root, Index Calculus
4. Artin's Conjecture, Discrete Log, Index Calculus
5. Quardratic Residues, Quardratic Non-Residues, Legendre Symbol
6. Gauss Lemma, Quadratic Reciprocity, Jacobi Symbol
7. Square Root, Tonelli's Algorithm, Cyclotomic Polynomials

Arithmetic Functions

1. Perfect Number, Arithmetic Function, List of Arithmetic Functions
2. Multiplicative, Completely Multiplicative, Convolution, Lehmer's Conjecture
3. Mobius μ Function, Mobius Inversion Formula
4. Mertens Conjecture, Riemann Hypothesis, ζ Functions, Riemann ζ Function

More Topics

1. Continued Fractions, Convergent, Inequalities
2. Quadratic Irrationalities, Pell's Equation, Four Squares Theorem
3. Pythagorean Triples, Fermat Descent, Rational Points on Conics