Prime Numbers (Theorems and the Infinitude of Primes) [Video]

by Dave

We discuss the difference between composites and prime numbers and then show how to find prime numbers using Eratosthenes’ sieve. We then prove the infinitude of prime numbers and show that every natural number has a prime divisor by assuming the well-ordering axiom.

In this video, I explain the difference between prime numbers and composites. Then I detail a method for finding prime numbers that remains effective today (for small prime numbers) even though Eratosthenes discovered this method thousands of years ago. After that, I consider prime divisors and show that every integer has a prime divisor. Finally, I show you how to prove that there are infinitely many primes.

Hi everyone, welcome back to my channel, I’m Dave. 

In today’s episode, I discuss prime numbers and some fascinating questions about them. In fact, towards the end, I will show you how to write one of the most beautiful proofs in all of mathematics.

Before you continue, you may want to checkout the video notes here Prime Numbers. Are you ready for some elementary number theory? You may also be interested to know that this video is part of the Number Theory Series playlist.

Prime Numbers (Theorems and the Infinitude of Primes)

This video is for anyone looking for an introduction to a deeper understanding of prime numbers.

How-to Learn Prime Numbers

After studying this video, you will know how to prove that there are infinitely many primes.

Step 1: Understand what primes are and the many questions about them.

The study of primes has been around for several millennia. Their importance, relevance, and usefulness persistence to this day. These numbers continue to provide insights and offer creativity beyond belief. For example, finding a complete list of all special types of primes is challenging. There are primes such as Fibonacci Primes, Mersenne Primes, Twin Primes, etc. And some people hold world records for finding a specific type of prime.

Step 2: Discover how to find prime numbers.

With the use of the Sieve of Eratosthenes, you learned a simple way to find primes and distinguish them from composites.

Step 3: Learn how to prove the two most basic facts about primes. 

These lemmas are extremely useful.

Step 4: Watch how to prove there are infinitely many primes. 

This proof is usually attributed to Euclid.

What is a prime number?

A prime number is a natural number greater than 1 that is divisible by no natural numbers other than 1 and itself. A natural number greater than 1 that is not prime is called a composite number.

How many prime numbers are there?

Prime Number Theorems Infinitude of Primes Collage of primary numbers from two to twenty three

Infamously, there are infinitely prime numbers, and this was proven by Euclid centuries ago. The proof usually goes like this 1) assume we have found all the prime numbers, say 2,3,5, …, P where P is the last prime number. 2) Then notice that the number N = 2*3*5*…*P +1 is not divisible by any prime number 2,3,5,…,P . But every number, including N must be divisible by some prime number. Therefore, we could not have found all prime numbers in the first place. 

What is the easiest way to find a prime number?

Using Eratosthenes’ sieve, one can write down all the natural numbers from 1 to n (wherever you want to stop). Then start crossing out multiplies of a number. Start by crossing out all multiples of 2, then cross out all multiples of 3, and so on until only primes are left. 

What is a prime divisor?

A theorem in elementary number theory states that every nonzero positive integer is divisible by some prime number. If a prime number divides an integer, then it is called a prime divisor of that integer. In this way, prime numbers are the building block of the integers.

In conclusion, I want to turn it over to you. 

Do you think the proof of infinitely many primes is beautiful?

So either way, let us know what you think in the comments for this video: Prime Numbers right now.