Introduction to Number Theory

Euclidean Algorithm, Linear Representation of GCD

In this lecture, you will be introduced to the basic concepts and statements required to study number theory. These include divisibility, primes, Euclidean algorithm, linear representation of the greatest common divisor. We advise you to pay attention to all new concepts, because they will meet frequently in the future.

Unique Factorization in Euclidean Rings

In this lecture, you will learn the most important property of Euclidean rings — uniqueness of prime factorization — and also learn how to apply it in practice to solve problems related to divisibility. As a beautiful application, the Fermat's Two Squares Theorem will be proved.

Euler's Function and it's Properties

In this lecture, you will be introduced to the very useful Euler's function, as well as some other arithmetic functions. We believe that you will find the Mobius Inversion Formula quite interesting. However pay special attention to the properties of the Euler's function including the "amusing property" as they will be used in one of the later lectures.

Modular Arithmetic Theorems

In this lecture, we will study tools for exploring situations in which there is no divisibility. You will learn what modular congruence is, as well as the basic theorems of modular arithmetic: Fermat's Little Theorem, Euler's Theorem, Wilson's Theorem.

Primitive Roots and Quadratic Remainders

In the first half of this lecture, you will be introduced to a powerful tool for solving problems modulo prime numbers: the primitive root. In the second half of the lecture, we will consider the theory of quadratic remainders which are needed to solve quadratic modular equations.

Continuants and Continued Fractions

This lecture is the first of two dedicated to continued fractions. You will get acquainted with continued fractions and learn how to decompose real numbers into continued fractions, as well as find the values of continued fractions. In addition, with the help of special polynomials in several variables (so called "continuants"), some of the properties of continued fractions will be proved, and these properties will be needed for rational approximations in the next lecture.

Convergence of Continued Fractions, Diophantine Approximations

This lecture is the second of two dedicated to continued fractions. You will find out why each continued fraction corresponds to a specific real number. In addition, you will learn how to find the best rational approximation of an irrational number based on the properties of continued fractions which will explain Archimedes' approximation for PI and the size of A4 paper.

Blichfeldt's Lemma, Minkowski's Lemma

This lecture has a combinatorial flavor just like the next one. You will learn useful theorems concerning figures on an integer lattice (i.e., a coordinate plane with an integer grid), and also learn how to apply these theorems and ideas in practice. We believe that you will love the artful proof of Fermat's Two Squares Theorem based on Minkowski's Lemma. Nevertheless, pay special attention to Minkowski's theorem: we will use it once again in the last lecture.

Kronecker's Theorem, Weyl's Theorem

This lecture has a combinatorial flavor just like the previous one. You will study the properties of the fractional parts of numbers of the form nα‎, where n is an arbitrary positive integer and α‎ is a fixed irrational number. Using the Kronecker's and Weyl's theorems, you will learn to find answers to unusual questions, for example, how often does the power of a positive integer begin with a given combination of digits, or what are the bounds of sets of values of trigonometric functions of integers.

Linear Diophantine Equations, Chinese Remainders Theorem

This lecture initiates a set of three lectures dedicated to Diophantine equations (this means equations that need to be solved in integers). In the first part of the lecture, you will learn how to solve linear equations in integers, as well as separately in non-negative integers (Sylvester's theorem). In the second part of the lecture, you will get acquainted with an important tool for simultaneous exploration of remainders modulo several different integers — the Chinese Remainder Theorem.

Nonlinear Diophantine Equations

In this lecture, you will learn various techniques that are useful for solving arbitrary equations in integers: the remainder method, the decomposition method, and the descent method. In addition, you will get acquainted with Pythagorean triples (the lengths of the sides of integer right-angled triangles), the notion of perfect numbers, Fermat's primes and Catalan's conjecture.

Quadratic Irrationalities and Pell's Equations

In this lecture, you will study the tools for working with quadratic irrationalities and consequently learn how to find all solutions of one of the brightest Diophantine equations — the Pell's equation. In the end of the lecture you will find an artful proof of existence of non-trivial solutions.