1.1. Integer Division

• Integer Division
• Greatest Common Divisor
• The Division Algorithm for Positive Integers
• The Euclidean Algorithm
• Linear Diophantine Equations
• Solutions to Linear Diophantine Equations

1.2. Prime Numbers, Composite Numbers, and Unique Factorization

• Primes Numbers
• Composite Numbers
• Unique Factorization
• Prime Power Decomposition
• Example Questions Regarding Primes and Composites

2.1. Congruent Integers Modulo m

• Congruences Modulo m
• Basic Properties of Congruences Modulo m 1
• Basic Properties of Congruences Modulo m 2
• Example Problems Regarding Congruences
• Tests for Divisibility
• Determining the Last Digit of Large Numbers

2.2. Linear Congruences

• Linear Congruences
• Existence of Solutions to Linear Congruences
• Solving Linear Congruences 1
• Solving Linear Congruences 2
• Solving Linear Congruences 3
• Solving Linear Diophantine Equations with Linear Congruences
• Systems of Linear Congruences
• The Chinese Remainder Theorem
• Solving Systems of Linear Congruences 1
• Solving Systems of Linear Congruences 2

2.3. Hensel's Lemma

• Hensel's Lemma
• Examples of Applying Hensel's Lemma
• Solving f(x) ≡ 0 (mod p^k) without Hensel's Lemma

2.4. RSA Encryption

• RSA Encryption
• Additional Examples of Creating RSA Decryption Keys
• Additional Examples of Finding RSA Decryption Keys
• Examples of Encrypting with RSA $C \equiv P^e \pmod {n}$
• Examples of Decrypting with RSA $P \equiv C^d \pmod {n}$

2.5. Fermat's Little Theorem and Wilson's Theorem

• Fermat's Little Theorem
• Wilson's Theorem
• Determining the Remainder of Large Factorials

3.1. The d Function

• The Number of Positive Divisors of an Integer ( Examples 1 | Examples 2 )

3.2. The σ Function

• The Sum of Positive Divisors of an Integer ( Examples 1 )
• k-Perfect, Abundant, and Deficient Numbers ( Examples 1 )
• Aliquot Sequences, Amicable Pairs, and Sociable Numbers ( Examples 1 )

3.3. Euler's ϕ Function

• Euler's Totient Function ( Examples 1 | Examples 2 )
• Euler's Totient Function on Odd and Even Natural Numbers
• Euler's Totient Theorem

4.1. The Order of n (mod m)

• The Order of a Natural Number Modulo m
• The Primitive Roots of a Natural Number
• Determining the Number of Primitive Roots a Prime Has
• Determining if r is a Primitive Root of n
• Finding Other Primitive Roots (mod p)

4.2. Quadratic Congruences

• Quadratic Congruences
• Euler's Criterion

4.3. Legendre Symbols

• Legendre Symbols
• Gauss's Lemma
  • Example Questions Regarding Gauss's Lemma
• Legendre Symbol Rules for (-1/p) and (2/p)
• Legendre Symbol Rules for (3/p) and (6/p)
• The Quadratic Reciprocity Theorem
  • Additional Examples of Evaluating Legendre Symbols

4.4. Jacobi Symbols

• Jacobi Symbols
• Examples of Computing Jacobi Symbols

4.5. Pythagorean Triangles

• Pythagorean Triangles
• Examples of Pythagorean Triangle Questions

5.1. Binary Quadratic Forms

• Binary Quadratic Forms
• The Discriminant of a Binary Quadratic Form
• Indefinite, Semidefinite, and Definite Binary Quadratic Forms
• IFF Criterion for a Binary Quadratic Form to Have a Discriminant d

5.2. Sums of Two Squares

• A Product of a Sum of Two Squares is a Sum of Two Squares
• Expressing Integers as a Sum of Two Squares
• Examples of Expressing Integers as a Sum of Two Squares

5.3. Equivalence and Reduction of Binary Quadratic Forms

• Proper Representation Criterion for Binary Quadratic Forms
• Equivalent Binary Quadratic Forms
• Example Problems Regarding Equivalent Binary Quadratic Forms
• The Class Number H(d) of a Discriminant d
• Reduced Binary Quadratic Forms
• Reducing Binary Quadratic Forms
• Examples of Reducing Binary Quadratic Forms 1
• Examples of Reducing Binary Quadratic Forms 2
• Computing the Class Number H(-3)
• Computing the Class Number H(-11)
• Computing the Class Number H(-39)
• Expressing Integers Properly as a Sum of Two Squares
• The Functions R(n), r(n), P(n), and N(n)
• A Formula for P(n) = N(n)

6.1. Arithmetic Functions

• Arithmetic Functions (Additive and Multiplicative)

6.2. Dirichlet Convolutions

• The Dirichlet Convolution of Two Arithmetic Functions
• Examples of Dirichlet Convolutions of Two Arithmetic Functions

6.3. The Möbius Inversion Formula

• The Möbius Function
• The Dirichlet Convolution μ * 1 = ꙇ
• The Möbius Inversion Formula

7.1. Farey Sequences and Dirichlet's Approximation Theorem

• Farey Sequences
• Theorems Regarding Farey Sequences
• Dirichlet's Approximation Theorem
• Dirichlet's Approximation Theorem by Farey Sequences

7.2. Irrational Number Approximation

• Irrational Number Approximation by Rational Numbers
• Hurwitz's Theorem
• An Example of a Series Converging to an Irrational Numbers

7.3. Algebraic Number Approximation

• Algebraic and Transcendental Numbers
• Algebraic Number Approximation
• An Example of a Series Converging to a Transcendental Number

7.4. Irrational Numbers

• The Rational Roots Theorem
• Rational and Irrational Values of cosθ, sinθ, and tanθ
• Proof that e is Irrational
• Proof that Pi is Irrational

7.5. Geometry of Numbers

• Blichfeldt's Principle
• Lattices in Rn
• Minkowski's Convex Body Theorem
• Proof that if p ≡ 1 (mod 4) then p is the Sum of 2 Squares by Minkowski's Convex Body Theorem
• Lagrange's Four Square Theorem

7.6. Continued Fractions

• Continued Fractions
• Finite Continued Fractions and Rational Numbers
• The Sequences (hn) and (kn)
• The nth Convergent of an Infinite Continued Fraction
• Convergence of Infinite Continued Fractions
• Infinite Continued Fractions and Irrational Numbers
• Examples of Computing Infinite Continued Fractions
• Irrational Number Approximation by Infinite Continued Fractions
• Criterion for a/b to be a Convergent of an Infinite Continued Fraction
• Periodic and Purely Periodic Continued Fractions
• IFF Criterion for an Irrational Number to be Periodic
• Periodic Continued Fractions are Quadratic Irrationals
• Quadratic Irrationals Have Periodic Continued Fractions
• IFF Criterion for an Irrational Number to be Purely Periodic

7.7. Pell's Equation

• Pell's Equation
• Solutions to x^2 - dy^2 = 1 and x^2 - dy^2 = -1
• Solutions to x^2 - dy^2 = N

8.1. Dirichlet Series

• The Prime Counting Function
• The von-Mangoldt Function
• The Chebyshev Functions
• Dirichlet Series
• The Riemann Zeta Function
• The Euler Product of a Dirichlet Series
• The Dirichlet Series for the von-Mangoldt Function