Elementary Number Theory
Math 304 — Fall 2025
- Meeting time and place: Tue/Thu 9:00 - 10:15, (Location TBA)
- Instructor: Egbert Rijke (erijke1[at]jh.edu)
- Office: Krieger Hall 213
- Office Hours: Wed 12:00 - 13:15, or by appointment
Course Description
This course offers an introduction to elementary number theory with minimal background prerequisites. We will be using Silverman’s A Friendly Introduction to Number Theory, and course notes written by the instructor. The course will cover essential concepts and some of the most celebrated results in elementary number theory, divisibility, the theorems of Fermat, Euler, and Wilson, the Chinese remainder theorem, prime numbers and factorization, some arithmetic functions, primitive roots, quadratic reciprocity, and Diophantine equations. Time permitting, additional topics from later chapters in the book, such as Pell’s equation, continued fractions, or factorization in the Gaussian integers, may also be included.
Course material
We will roughly cover the material of Chapters 1 through 36 of the book A Friendly Introduction to Number Theory by Joseph Silverman. Course notes will be provided by the instructor.
Provisional List of Topics
- The Nature of Numbers
- The Foundational Debates of the 19th and 20th Centuries
- Induction and Recursion
- Arithmetic of Natural Numbers
- The Numbers of Fibonacci and Lucas
- Finite Sums and Products
- Counting
- The Standard Finite Sets
- Counting Bijections
- Counting Subsets
- The Binomial Theorem
- The Inclusion-Exclusion Principle
- Exercises (19)
- The Integers
- Cyclically Ordered Sets
- A Structural Definition of the Integers
- Integer Arithmetic
- Euclidean Division and Representability
- The Well-Ordering Principle of the Natural Numbers
- Euclidean Division
- The Representability Theorem
- Linear Diophantine Equations
- Divisibility
- Ideals of Integers
- The Ordering by Divisibility
- Greatest Common Divisors
- Euclid’s Algorithm
- Linear Diophantine Equations in Multiple Variables
- Prime Numbers
- The Fundamental Theorem of Arithmetic
- The Infinitude of Primes
- Legendre’s Formula and Kummer’s Theorem
- Bertrand’s Postulate
- The Rational Numbers
- Integer Fractions
- Farey Fractions
- A Structural Definition of The Rational Numbers
- The Stern–Brocot Tree
- Finite Simple Continued Fractions
- Polynomials
- Polynomials with Integer Coefficients
- Lagrange’s Interpolation Theorem
- Fixed Divisors of Integer Polynomials
- Pythagorean Triples
- The Pythagorean Theorem
- Euclid’s Parametrization of the Pythagorean Triples
- Rational Points on the Unit Circle
- The Tree of Primitive Pythagorean Triples
- Conics
- Infinite Descent
- The Method of Infinite Descent
- The Unsolvability of x^4+y^4=z^4
- Vieta Jumping
- Pell’s Equation
- Congruences
- The Congruence Relations
- Equivalence Relations
- Equivalence Classes and Residue Systems
- Reduced Residue Systems
- Modular Arithmetic
- The Integers Modulo n
- Solving Linear Congruences
- Fermat’s Little Theorem
- Euler’s Theorem
- Wilson’s Theorem
- Lucas’s Theorem
- The Quadratic Character of -1
- Fermat’s Two-Square Theorem
- Reduction to the Primes
- Euler’s Proof by Infinite Descent
- Zagier’s Proof Using Fixed Points of Involutions
- The Method of Brillhart–Hermite–Serret
- The Gaussian Integers
- Definition and Basic Properties of the Gaussian Integers
- The Greatest Common Divisor of Gaussian Integers
- Primes in the Gaussian Integers
- Fermat’s Two-Square Theorem
- Systems of Linear Congruences
- Solving Multiple Linear Congruences Simultaneously
- The Chinese Remainder Theorem
- Linear Congruences in Multiple Variables
- Multiplicativity of Euler’s Totient Function
- Polynomial Congruences
- Polynomial Congruences of Prime Moduli
- Polynomial Congruences of Composite Moduli
- Reduced Polynomials Modulo a Prime
- The Elementary Symmetric Polynomials
- Primitive Roots
- The Multiplicative Order of an Integer Modulo n
- The Infinitude of Primes Congruent to 1 Modulo Powers of 2
- Counting Elements of a Given Order Modulo a Prime
- Primitive Roots
- Quadratic Residues
- Quadratic Congruences
- Quadratic Residues
- Legendre Symbols
- Euler’s Criterion
- Euler’s Prime-Generating Polynomial
- Quadratic Reciprocity
- The Quadratic Character of 2
- The Statement of Quadratic Reciprocity
- Gauss’s Lemma
- Eisenstein’s Proof
- Arithmetic Functions
- Multiplicative Functions
- Dirichlet Convolution
- The Möbius Inversion Formula
- Dirichlet Inverses
- The Distribution of the Prime Numbers
- The Bachmann–Landau Notation for Asymptotic Growth
- An Elementary Estimate of the Prime Counting Function
- Chebyshev’s Theorem
Literature
This course was originally designed to follow Andrews’ book Number Theory (Dover, 1994) fairly closely. One distinctive feature of this book is that it presents many of the most important theorems from two perspectives: a combinatorial one and an abstract one. This dual approach helps to clarify not only why these results are true but also appreciate why they are natural and inevitable within the broader framework of mathematics.
From the Fall semester of 2025 onwards, the course was redesigned to follow Silverman’s A Friendly Introduction to Number Theory (3rd ed., Pearson, 2006)]. Silverman’s book encourages the reader to explore numbers, hypothesize their properties and relations, and effectively teaches students how to generate ideas towards proving number theoretic theorems. It is written a very accessible, conversational style, with practical examples and exercises.
There are many further excellent sources to learn number theory from. One of my personal favorites for its clarity and accessibility is LeVeque’s Topics in Number Theory. Both Andrews’, Silverman’s, and LeVeque’s textbooks contain plenty of exercises, most of which are very fun.
The undisputed classic textbook on number theory, which is warmly recommended for any aspiring number theorist, is Hardy & Wright’s An Introduction to the Theory of Numbers (6th ed., Oxford Univ. Press, 2008). This book covers all the essential topics in number theory, including elementary number theory and analytical number theory. It is more comprehensive and also provides more historical notes. The textbook of Hardy and Wright does not provide exercises, but it contains the proofs of many important facts in number theory that are stated as exercises elsewhere.
Online resources
There are plenty of ways to learn number theory and engage with communities of mathematicians and students online. First and foremost, Wikipedia has many excellent pages on topics from and related to number theory. The website Math.StackExchange is dedicated to answering any kind of mathematical question, although more research-oriented questions are usually posed on MathOverflow. The website Art of Problem Solving is dedicated to contest mathematics and the problem-solving techniques necessary to do well in competitions such as the International Mathematical Olympiad.
Furthermore, there are some popular channels on video-sharing sites such as YouTube, Twitch, and TikTok. We mention some of the most notable:
- Lectures:
- Richard Borcherds (Fields Medalist) has recorded many of his Berkeley number-theory lectures in the Introduction to Number Theory playlist on YouTube.
- Problem solving:
- Michael Penn has an excellent YouTube channel where he solves mathematical problems on a blackboard at roughly the level of this course.
- vEnhance (Evan Chen) is an IMO gold medalist and the author of the wonderful book Euclidean Geometry in Mathematical Olympiads \cite{chen2016}. He streams live solves of Olympiad math problems on Twitch, and his videos are also available on YouTube.
- OmegaLearn is Jonathan Huang’s YouTube channel focused on solving IMO problems. Some of his playlists dive deeper into specific techniques, such as Fermat’s method of infinite descent.
- Blackpenredpen is Steve Chow’s problem-solving channel, covering topics in calculus, algebra, and number theory.
- Mathematics for a broad audience:
- 3Blue1Brown explores a variety of topics related to computer science and mathematics using compelling visualizations.
- Mathologer is Burkard Polster’s YouTube channel for recreational mathematics. Many of his videos contain elegant visual proofs; his videos on Fermat’s Two-Square Theorem and the Quadratic Reciprocity Theorem are especially worth watching.
- Numberphile is a long-running YouTube channel by Brady Haran, featuring mathematicians who explain a variety of mathematical phenomena, including topics in number theory. A closely related YouTube channel is Tom Rocks Maths, by Tom Crawford.
- PeakMath has an excellent series on the Riemann Hypothesis, called the Riemann Hypothesis Saga, in which they topics such as the Langlands program and the Birch–Swinnerton Dyer conjecture are made accessible.
You might also enjoy joining some Discord servers, such as the Art of Problem Solving (AoPS) Community Server, the Math Stack Exchange Discord, or the OmegaLearn Server. If you are interested in formalization of mathematics, using proof assistants such as Agda, Lean, or Rocq, you’ll also find thriving online communities that are focused on building large libraries of formalized mathematics. Lean’s Natural Numbers Game is especially worth trying, especially if you are new to induction and recursion. This set of lecture notes is currently being formalized in the agda-unimath library.
Finally, you might find generative AI tools such as ChatGPT useful in exploring mathematical topics. These tools are helpful because you can ask them to explain any topic that interests you, and the quality of their answers is quickly improving with each new version. Be careful, however, to ensure you understand the answers for yourself. Large language models are prone to give overconfident presentations, and as such AI-generated reasoning is not always reliable.
Some Complementary Reading