MATH 110: Introduction to Number Theory

Lecture Notes

• Lecture 0

Linear Diophantine Equations

• Lecture 1.1 (Euclidean) Division Algorithm.
• Lecture 1.2 (Binary) linear Diophantine equation
• Lecture 1.3 Homogeneous linear Diophantine equations
• Lecture 1.4 Solution set of binary linear Diophantine equations
• Supplementary Notes

Prime Numbers

• Lecture 2.1 Hasse diagram
• Lecture 2.2a Prime Factorization
• Lecture 2.2b Prime Factorization
• Lecture 2.3 Translation between two worlds
• Lecture 2.4 Infinitude of primes
• Lecture 2.5 Distribution of Primes
• Lecture 2.6 Divisor set and Multiplicative functions
• Lecture 2.7 General sigma functions
• Lecture 2.8 Euclid-Euler theorem
• Supplementary Notes

Rational and Algebraic Numbers

• Lecture 3.1 Rational numbers
• Lecture 3.2 Algebraic numbers
• Lecture 3.3 Diophantine approximation
• Lecture 3.4 Ford circles
• Lecture 3.5 Mediant
• Lecture 3.6 Farey sequence
• Lecture 3.7 Dirichlet’s approximation theorem
• Lecture 3.8 Higher Diophantine equations
• Lecture 3.9 Higher Diophantine equations
• Supplementary Notes

Modular World and Modular Dynamics

• Lecture 4.1 Congruence and modulus
• Lecture 4.2 Modular arithmetic
• Lecture 4.3 Modular dynamic
• Lecture 4.4 Multiplicative modular dynamic
• Lecture 4.5 Primality testing
• Lecture 4.6 Relation between additive and multiplicative dynamics
• Lecture 4.7 Discrete logarithm
• Lecture 4.8 Application to cryptography
• Lecture 4.9 Primitive root theorem
• Supplementary Notes

Modular Polynomials

• Lecture 5.1 Modular polynomials
• Lecture 5.2 Division of modular polynomials
• Lecture 5.3 (Euclidean) division algorithm and Prime factorization
• Lecture 5.4a Roots and degree
• Lecture 5.4b Finishing proving Primitive Root Theorem
• Supplementary Notes

Assembling Modular Worlds

• Lecture 6.1 (binary) Chinese Remainder Theorem
• Lecture 6.2 (multivariable) Chinese Remainder Theorem
• Lecture 6.3 Chinese Remainder Theorem: Applications
• Lecture 6.4 Reduction and Lifting
• Lecture 6.5 Hensel’s Lifting Theorem
• Supplementary Notes