1. Natural Numbers, Integers, Rational Numbers

2. Divisibility and Division Algorithm

3. Prime and Composite Numbers

4. Factorization and Unique Factorization Theorem (Fundamental Theorem of Arithmetic)

5. Even and Odd Numbers

Modular Arithmetic 

1. Congruence Relations

2. Properties of Modular Arithmetic

3. Modular Inverses

4. Systems of Congruences

5. Chinese Remainder Theorem

6. Residue Classes

Theorems in Number Theory 

1. Fermat’s Little Theorem

2. Euler’s Theorem

3. Wilson’s Theorem

4. Lagrange's Theorem (in modular arithmetic)

Arithmetic Functions 

1. Definition of Arithmetic Functions

2. Multiplicative and Completely Multiplicative Functions

3. Möbius Function μ(n)

4. Divisor Function σ(n)

5. Number of Divisors Function d(n)

Diophantine Equations 

1. Linear Diophantine Equations

2. Pythagorean Triples

3. Pell’s Equation

4. Integer Solutions of Polynomial Equations

Advanced Topics (Optional for Undergrad) 

1. Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

2. Euclidean Algorithm and Extended Euclidean Algorithm

3. Linear Diophantine Equations

4. Bézout’s Identity

Prime Numbers and Their Properties 

1. Sieve of Eratosthenes

2. Prime Factorization

3. Distribution of Primes

4. Infinitude of Primes

5. Fermat Numbers and Mersenne Primes

Euler’s Totient Function 

1. Definition and Calculation of ϕ(n)

2. Properties of ϕ(n)

3. Euler’s Theorem and its Applications

Quadratic Residues 

1. Legendre Symbol

2. Euler’s Criterion

3. Quadratic Reciprocity Law

4. Solving Quadratic Congruences

Continued Fractions