Elementary Number Theory (in English).
During the Summer of 2022, I offered a basic course on Topics in Elementary Number Theory for first-year high school students, in response to a request for preparing the Chinese National High School Mathematics League in 2023. At the end of the course, for the convenience of the AMC/AIME/AHSME/USAMO participants, the lecture notes will be translated into English, collated into a number of documents, and gradually uploaded onto this page.
- Reference: Lecture Notes of Mr. Tao Han. (I apologize for not mentioning the provenance of exercise problems.)
The Notes Directory
- On minimal prime divisors.
Trick introduction; some basic examples and advanced problems (IMO 2020 involved).
- Fermat’s little theorem.
Statement and proof; primary applications (finding the remainder of a large number, Carmichael numbers, division problems); two more advanced problems (IMO 2005 involved).
- Euler’s theorem.
Statement and proof; primary applications; two difficult problems (Eulerian sum equation, triangle dual product).
- Order theory and primitive root.
Basic notions, features of primitive root and order; a crash application to Mersenne integers.
- Prime power congruence and Hensel’s lemma.
Congruence lifting, Hensel’s 1st and 2nd lemmas, applications of Hensel’s lemma on solving congruence equations.
- Pell equations.
An introduction to Pell equations, the structure of solutions, type II Pell equations; problems and examples (IMO 2001 Shortlist involved).
- Lifting-the-exponent lemma.
Statements and proofs (for odd and even integers); some typical applications on solving Chinese GMO/TST problems.
Exercise Problems
- Problem Set 1 (Hints, Solution).
Euclidean algorithm, elementary computations on gcds, the prime-power-order argument.
- Problem Set 2 (Hints, Solution).
Prime number theorem, p-adic expansions and valuations, Kummer’s theorem, lifting-the-exponent lemma (optional), division problems about combinatorial numbers.