Fundamental Theorem of Arithmetic Infinity of Prime Numbers

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Элементарная Математика

Published on Feb 12, 2022
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In this lecture, we will continue the Number Theory direction, which began with two lectures. In the first, we got acquainted with the greatest common divisor and the least common multiple, and also learned how to find the GCD using the Euclidean algorithm. You can watch the lecture at the link • Least common multiple (LCM) and la... . In the second lecture, we considered the fundamental theorem on the greatest common divisor and solved the problem of solvability of a linear equation with several variables in integers. You can watch the second lecture at the link • The fundamental theorem on the greatest common d... Today we will turn to the fundamental theorem of arithmetic and prove it. In addition, we will consider the canonical decomposition of a natural number, which easily allows us to obtain all the divisors of the number. We will analyze a simple example and find all the divisors of the number 120. As an exercise, you will be asked to find the number of all divisors of a number and the sum of all divisors of a number in general form, which is easy to do after the example analyzed. And so that the lecture is not too short, we will also prove the infinity of the set of prime numbers, giving two simple proofs. All these statements will be proven with the help of one lemma, which unites the results, and is very simple and practically obvious. read by Igor Tinyakov #elementarymathematics #basictheoremofarithmetic #primenumbers #compositenumbers

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