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# Prime Numbers

Richard Patterson on Flickr

Prime numbers are one of those fun little things in mathematics that start so simple but end up being so important, not just to math but to programming as well! By the end of this article we’re going to explain why certain numbers are so vitally important to talking, shopping, and existing online safely.

So first let’s review what prime numbers are. They’re integers, whole numbers, other than 1 that can only be evenly divided by themselves or by 1. What I mean by evenly divided is giving a whole number as a result. So 6/2 evenly divides into 3, but 5/2 doesn’t because 5/2 is 2.5.

Some really quick examples: 2 is prime, 3 is prime, but 4 isn’t because 4/2 is 2. The reason why mathematicians and computer scientists alike care about prime numbers is that every whole number is either a prime number or can be represented uniquely as a product of prime numbers. What “uniquely” means here is that there aren’t choices to make, there’s only one way to break the number into a product of primes.

As an example, what about a number like 236? 236 ends up being the product 2*2*59, all of which are prime numbers. Wait, you might ask, how do you know that 59 is a prime number? It’s true, that doesn’t seem incredibly obvious does it? To be able to break a number into primes we need to know which numbers are prime in the first place. We’ll come back to that question shortly after we discuss why computer scientists use prime numbers.

What’s the importance of prime numbers for online life? That comes down to cryptography and specifically something called asymmetric encryption.

Asymmetric encryption is generally not used to make big swaths of data secure but rather is an important part of handshakes, the kinds of protocols used to establish that each computer in a conversation is exactly who it says it is. In asymmetric encryption, you have a public key and a private key. The public key is public, meaning anyone can have it and in fact needs to have it for the next part. The private key? You alone should have your private key. To prove your identity, you’re going to encrypt a message with your private key and then if it can be unencrypted with your public key you’ve proven you sent the message.

To generate your public and private key, you first pick two very large prime numbers: you multiply them together to make your public key and do a different, more complex, set of operations on them to make the private key. The entire reason why you can distribute the public key freely is because while it’s very easy to multiply two numbers, even big ones, it’s incredibly hard to look at a huge number—like numbers with hundreds of digits huge—and be able to figure out what prime numbers make them up.

Here as an example, try to figure out which two prime numbers make up this relatively small number: 3183379. At the end of the article I’ll tell you which they were!

This is such a hard problem that there’s only one kind of computer that could solve it quickly and that’s a quantum computer. This is actually one of the main things that makes people excited and scared of quantum computers: they can be used to break asymmetric encryption.

So back to the question of how do you know a number is prime? It turns out, this is a pretty hard question because the only way to know a number is prime is because you’ve failed to show it isn’t. That’s because the definition of “prime” means you’ve shown there aren’t any other numbers that divide it. The simplest solution in code is the brute force check, where you check one by one to see if there is a number that can divide the number you’re testing! If there isn’t one, then you know the number is prime.

Here’s how a brute force check might work with code by first importing a library with math functions:

``` import math
def isprime(n):
isPrime = True # assume it's prime until you find a factor
for x in range(2,1+int(math.sqrt(n))):
# check from 2 til squareroot of n
# (the 1 + is so the for loop includes the sqrt)
if n % x == 0:
isPrime = False
return isPrime
```

But a more sophisticated, and actually quite old, algorithm is the sieve of Eratosthenes which is best described like this: if you want to find all the primes below a certain number, say 100, then first start by writing down each number below 100. Now go to the first prime number, 2, and cross off all multiples of 2 starting with 4. The next number not crossed off is 3, so cross off all multiples of 3 that haven’t already been crossed off. The next number not crossed off will be 5, so continue the process until there’s no more numbers under 100 to cross off. What isn’t crossed off are all the prime numbers below your limit.

The Sieve of Eratosthenes is an algorithm that has more upfront work but is actually more efficient because it calculates not just whether one number is prime but what all the prime numbers below a certain number are.

The funny thing, though, is that in industrial cryptographic applications things like this aren’t used but rather probabilistic tests, which rather than giving an absolutely correct answer give an answer that’s very unlikely to be wrong. In the asymmetric encryption we talked about, they generally use something called Fermat’s test which is incredibly fast and has a ridiculously tiny chance of being wrong. This allows for fast public/private key generation!

So prime numbers are both an interesting thing to study in their own right but also weirdly important to the ways we communicate online every day. Sometimes the most unlikely parts of mathematics suddenly become the most practical!

### Eratosthenes

While it seems people from hundreds and thousands of years ago had little impact on our world of phones, computers, cars, and other amazing tools, the truth is our digital world relies heavily on inventions and people from the distant past.

Eratosthenes is one such inventor. Born around 276 BCE in a Greek colony called Cyrene, today known as Shahhat in Libya, he used mathematics to calculate the circumference of the earth based upon the angle of sunlight at different locations and points in time along the coast of North Africa. He used trigonometry to measure distances with longitude and latitude. To identify prime numbers, he also developed an algorithm called the Sieve of Eratosthenes, which is described in our article about prime numbers and encryption, to the left of this article.

However, math was only one field in which Eratosthenes excelled.  He also was a noted geographer, poet, historian, philosopher, astronomer, and music theorist. Eratosthenes also discovered that a year was slightly more than 365 days and thus created the leap year, a day added to a calendar every four years. He also created a map of 675 stars as well as measured the distance between the earth and sun. The Egyptian king Ptolemy III invited him to run the Great Library of Alexandria which at the time held originals and copies of most books published in the ancient world.

Perhaps the most amazing detail about Eratosthenes, and others from the ancient world, they used sticks and camels and other primitive tools to discover useful things we use today in our modern world.

#### Eratothenes bio

https://www.thoughtco.com/eratosthenes-biography-1435011

https://www.famousscientists.org/eratosthenes/

#### Eratothenes measures the earth

https://www.aps.org/publications/apsnews/200606/history.cfm

#### Eratothenes map

https://digitalmapsoftheancientworld.com/ancient-maps/eratosthenes-map/

#### Eratothenes facts and worksheets

https://kidskonnect.com/people/eratosthenes/

#### Sieve of eratosthenes

https://www.bubblyprimes.com/sieve-of-eratosthenes/

#### What are prime numbers?

https://en.wikipedia.org/wiki/Prime_number

#### Prime numbers chart and caculators

https://www.mathsisfun.com/prime_numbers.html

#### Why prime numbers are so exciting

https://kids.frontiersin.org/article/10.3389/frym.2018.00040

#### Prime number facts

https://www.factmonster.com/math-science/mathematics/prime-numbers-facts-examples-table-of-all-up-to-1000

#### Checking if a number is prime

http://www.counton.org/explorer/primes/checking-if-a-number-is-prime/

#### Finding prime numbers in Python

https://www.programiz.com/python-programming/examples/prime-number

#### Why are prime numbers important?

https://www.extremetech.com/extreme/219570-what-are-prime-numbers-and-why-are-they-so-vital-to-modern-life

#### Introduction to prime numbers

http://njlovesmath.com/an-introduction-to-prime-numbers/

#### fun prime number facts

https://facts.net/science/prime-numbers-facts

#### Prime number explanation for parents and kids

https://thirdspacelearning.com/blog/what-is-a-prime-number/

#### Prime factorization puzzles

https://www.teacherspayteachers.com/Browse/Search:prime%20factorization%20puzzle

#### Prime number maze

https://www.worksheetworks.com/math/numbers/prime-number-maze.html

#### Prime numbers game

https://www.education.com/activity/article/slap-happy/