### Number Theory

### Introduction

- Number theory
- Mathematicians
- Diophantine Equations

### Natural Number and Integer

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

### Modular Arithmetic

- Congruence, Addition, Subtraction, Multiplication, Division (ModInverse)
- Complete Residue System, Reduced Residue System, Euler's Totient Function
- Euler's Theorem, Fermat's Little Theorem
- Inverse of Elements, Division, Wilson's Theorem
- Congruence Equation, Linear Congruence Equation
- Linear Congruences, Chinese Remainder Theorem (CRT)

### Prime Power and Primitive Root

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

### Arithmetic Functions

- Perfect Number, Arithmetic Function, List of Arithmetic Functions
- Multiplicative, Completely Multiplicative, Convolution, Lehmer's Conjecture
- Mobius
*μ* Function, Mobius Inversion Formula
- Mertens Conjecture, Riemann Hypothesis,
*ζ* Functions, Riemann *ζ* Function

### More Topics

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