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 download the video notes here: Prime Numbers pdf notes. 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)

## FAQ Prime Numbers

## 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?

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.

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