Number Theory

Number Theory Curriculum

This curriculum provides a structured overview of the study of number theory.

Foundations

Divisibility:

• The division algorithm
• Greatest common divisor (GCD)
• Euclidean algorithm
• Bézout's identity
• Least common multiple (LCM)

Prime Numbers:

• Definition and properties of prime numbers
• The Fundamental Theorem of Arithmetic
• Distribution of prime numbers
• The Prime Number Theorem
• Tests for primality

Congruences:

• Definition and properties of congruences
• Residue classes
• Linear congruences
• The Chinese Remainder Theorem
• Fermat's Little Theorem

Core Concepts

Multiplicative Functions:

• Definition and examples of multiplicative functions
• The Euler's totient function
• Dirichlet convolution
• The Möbius function and inversion formula

Quadratic Reciprocity:

• Quadratic residues and non-residues
• The Legendre symbol
• The Law of Quadratic Reciprocity
• Applications of quadratic reciprocity

Diophantine Equations:

• Linear Diophantine equations
• Pythagorean triples
• Fermat's Last Theorem (introduction)
• Elliptic curves (introduction)

Further Topics

Algebraic Number Theory:

• Number fields
• Algebraic integers
• Ideals and unique factorization
• The class group

Analytic Number Theory:

• Dirichlet series
• The Riemann Zeta function
• The Prime Number Theorem (proof)
• The distribution of primes in arithmetic progressions