Olympiad Number Theory Book

2026 Update: 4th Edition Released with over 500 Problems

Version 4.0 of the textbook is complete! Note: I do not want money to be a barrier to anyone accessing the book. If you need a discount code, please email me at ‘x’@numbertheoryguy.com (x = first name).

To stay up to date on future versions, please sign up for the mailing list. An AoPS forum for book discussion is available here.

I hope you sit back and enjoy the problems!

New Update in 2025: YouTube Videos to accompany the textbook

Once a week, a new video will be released on this page that supports the textbook. Be sure to subscribe to my YouTube channel so you don’t miss any of the updates! 

Sample Videos

We look at Euclid's Algorithm for quickly finding the greatest common divisor of numbers, with examples from Mandelbrot and AIME. Problems from Chapter 5 of my book: https://numbertheoryguy.com/publications/olympiad-number-theory-book/
How Did Euclid Quickly Find These GCD’s?
We show that the sum of cubes equals the square of the sum of the first n integers, using telescoping sums and the binomial theorem.
A Beautiful Algebraic Identity #maths #education #algebra
We explore the sum-of-cubes formula and prove the identity using telescoping sums and the binomial theorem. More from Chapters 1 & 2 of my number theory book: https://numbertheoryguy.com/publications/olympiad-number-theory-book/
Sum of Cubes: Telescoping Proof
Observe that 1^3+5^3+3^3=153. In what other bases does this hold?
Narcissistic Numbers #maths #education #numbertheory
Learn how to take the cube root of two-digit numbers cubed in your head!
Cube Root Party Trick
We see some fast divisibility-rule tricks for 9, 11, 91, and 99 with an HMMT problem. Solutions from Chapter 3: https://numbertheoryguy.com/publications/olympiad-number-theory-book/
Quick Divisibility Rule Tricks
We demonstrate the Ramsey Number R(3,3) is at most 6 in this video.
What Is True About a Party of Six People?
We explore a family of 3-digit Armstrong Numbers using base numbers. This video expands on Chapters 3 and 4 of my number theory book: https://numbertheoryguy.com/publications/olympiad-number-theory-book/
Armstrong Numbers in Infinite Bases
In this video, we find all n-digit numbers that can be written as the sum of the nth powers of their own digits, also known as Armstrong numbers or narcissistic numbers. This video combines ideas from algorithms, search, and number theory together.
Searching for Numbers in a 10⁶⁰ Haystack
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Olympiad Number Theory Through Challenging Problems

Third Edition (pdf) 

Third Edition published December 18th, 2016.

An 120 page introductory Olympiad Number Theory book I wrote for anyone with a passion for number theory and problem solving published on the Art of Problem Solving website. Each section begins by introducing a main concept or idea, and then contains many engaging and challenging problems. My goal for the text was to show how several problem solving skills–experimenting with small cases, simplification, and pattern finding–can be used to tackle a wide variety of Olympiad Number Theory problems. There are five chapters on Divisibility, Modular Arithmetic, p-adic valuation, Diophantine Equations, and Problem Solving Strategies.

Second Edition

(3/2/2015)  Olympiad Number Theory Through Challenging Problems (pdf) 

First Edition

(8/25/2013) The Basics of Olympiad Number Theory (pdf) (Problems and Theorem)