Основная теорема арифметики | Бесконечность множества простых чисел

In this lecture, we will continue the Number Theory section, which began with two lectures. In the first, we learned about the greatest common divisor and least common multiple, and learned how to find the greatest common divisor using the Euclidean algorithm. You can watch the lecture at    • Наименьшее общее кратное (НОК) и наибольши...   . In the second lecture, we examined the fundamental theorem of greatest common divisors and solved the problem of the solvability of a linear equation with several integer variables. You can watch the second lecture at    • Основная теорема о наибольшем общем делите...   Today, we will address the fundamental theorem of arithmetic and prove it. Additionally, we will consider the canonical factorization of a natural number, which easily yields all the divisors of the number. We will analyze a simple example and find all the divisors of 120. As an exercise, you will be asked to find the total number of divisors of the number and the sum of all the divisors of the number in general form, which is easy to do after analyzing the example. To make sure the lecture isn't too short, we'll also prove the infinity of the set of prime numbers by giving two simple proofs. All these statements will be proven using a single lemma that unifies the results and is very simple and practically obvious. Read by Igor Tinyakov #elementarymathematics #basictheoremofarithmetic #primes #compositenumbers