Sgibnev A.I. Divisibility of Integers and Prime Numbers

Moscow: Independent University of Moscow (IUM), 2022. — 127 p.In presenting the course “Divisibility and prime numbers”, there are two main approaches. The first approach focuses on the logic of the presentation: all statements are proved, and those that are not proved are not used. The second approach focuses on problems: the fundamental theorem of arithmetic is stated at the beginning and is given without proof, which makes it possible to avoid theoretical subtleties, and immediately proceed to solving meaningful problems. The author tried to use the middle way. It seemed to him fundamentally important in a mathematical course to prove all the statements sooner or later.Here is a brief outline for teachers and trained student:
The notion of prime number is introduced, and we provethat any number can be factored into primes; however, the question of uniqueness of this factorization is studied later (Lesson 4).
Euclid’s algorithm is introduced (Lesson 5).
Euclid’s algorithm is used to prove the fundamental lemma (Lesson 6).
The prime divisor theorem is deduced from the fundamental lemma (Lesson 7).
The prime divisor theorem is used to prove the uniqueness of factoring into primes (Lesson 8).
Also, in Lesson 6, the fundamental lemma is used to deduce the theorems on coprime divisors and factor cancellation (but they are optional in a minimal logical scheme).Lessons 1–4 are intended for students of intermediate forms, and Lessons 5–8 for students of upper forms.
The most important problems are marked with the “+” sign, and the most difficult problems are marked with the “∗” sign. Unless stated otherwise, “numbers” are understood as “integers”.Divisibility of integers.
Divisibility tests.
Division with remainder.
Prime numbers.
Common divisors and common multiples. Euclid’s Algorithm.
Diophantine equations.
Prime divisor theorem.
Factorization into primes. The Fundamental Theorem of Arithmetic.
Additional Problems.
Hints and Short Solutions.
Appendix 1. Two Unsolved Problems about Primes.
Appendix 2. Several Research Problems Related to Divisibility.
Appendix 3. An Alternative Outline of the Course.
Handout Material.
List of References and Web Resources.