I read that they have some uses within online encryption, but to be honest I never really thought about why we learned them in school until this morning.
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One of the key tenets that makes modern cryptography work is that [prime factorization](https://en.wikipedia.org/wiki/Integer_factorization) of a number takes a long time. Multiplying two large prime numbers together is very easy. But doing the reverse – taking a very large number and trying to figure out what two numbers you used to multiply them together – is difficult. This is leveraged in many cryptographic schemes like RSA.
They’re fundamental to RSA encryption, arguably one of the most important advances in digital security technology.
The principle, is that if you take two large prime numbers, the product can be computed super fast… But if I give you a a big number and ask what two primes made that number, it’s really hard… Furthermore, if I supply one prime and you supply one prime, I can get your prime by dividing the product by mine and vice versa.
Cybersecurity.
One way that computers stay safe is by requiring a number to access them. If another device says the number, the computer grants access. Like a password.
If you let another computer try enough times, it will multiply numbers until it achieves the computer code through complex math that doesn’t matter for this. Basically, if a number is a product of numbers, it can be cracked pretty easily.
Not so for prime numbers, since it’s not the product of numbers, you need to guess it point blank. This takes a longer time, and gives operators more time to stop the unauthorized access.
One core aspect of encryption is based on the idea that if you take a very large number (think 100 digits) it is very hard to figure out what numbers multiplied together get to that number (factoring). But it is super easy to take a bunch of numbers and multiply them to get a huge number. With prime numbers that process is unique. That is used a lot in encryption because you can share the giant number with anyone as an encryption key, but only the person that knows the original factors can prove it is the right key.
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Primes factorization is a really simple way of explaining the “one-way” processes that underpin encryption.
The property of primes you have to know is that every single number has only one so called “prime factorization” which just means that if you only multiply prime numbers, there is only one combination of those prime numbers that gives you your original number.
Example 1: 17*47=499. The only way you can get 499 from prime numbers is by multiplying 17 and 47.
Example 2: 2*2*2*19=152 is the only way to multiply prime numbers and get 152.
Etc. You can do that for every number.
The trick is that while it’s “easy” to multiply numbers together, it’s incredibly hard to reverse it. Like, try to find the prime factorization of 2455. (Actually don’t try, it’s not going to work, it’s 5*491)
You basically have to go through every prime number one by one, and multiply them to find out what it might be.
So you have an operation that’s easy in one direction and hard to reverse.
You have, a basic encryption algorithm.
There are thousands of applications of prime numbers otherwise, but this one is incredibly simple to explain despite being quite a powerful observation.
most common actual use case is credit card security/ bank account security.
The reason being, that you can multiply prime numbers really fast (like any other numbers multiplied are fast on calculators/computors) generating a security number (easily spoken).
However, if you have only that security number, but want to know out of which numbers it has been muliplied, computers need a lot of time, since divisions are more complex.
So it is easy if you know what the base (prime-)numbers are, but very hard if you want to find them. That makes them perfect for security.
And if you take large prime numbers or more of them that can be enough to increase the calculation time it takes a computer to crack it into years.
Prime numbers and numbers which are “co-prime” to each other are important in engineering, as they minimize linkage between different components.
For example, if you have moving parts next to each other which might cause a vibration, then having the two parts each moving at a different prime number speed, minimizes the vibration between parts. This concept has been used in jet engines and other high speed machinery with multiple moving parts.
It can also be used to minimize electrical interference between different wires. A common way in modern electronics to transmit a signal is to use a pair of wires which are twisted together. The electromagnetic field from one wire cancels out the other, and the twisting keeps the wires close together. However, if you have two twisted pairs directly next to each other, then if the twists line up, you can have areas where the electromagnetic field doesn’t cancel fully and one pair’s signal can leak into the other pair. If you use different twist rates, and use prime numbers for the number of twists, then you minimize the number of points where the twists line up, and keep interference to the minimum – this type of construction is used in modern twisted pair cables, such as ethernet cables.
There are some uses in computer security, as one of the first public-key encryption systems (the RSA cryptosystem) was based on prime numbers. Public-key cryptosystems are important, because they allow you to have make a secure channel, without pre-arranging it. You can convert any communications channel into a secure channel at any time.
A symmetric cryptosystem (e.g. a password) requires that you first agree a password with whoever you want to communicate securely with – so you need to get your password to the other party securely – so you need a secure communications channel to get the password to them. Oh wait.
The use of RSA public-key encryption has largely fallen out of favor these days for internet security, although it is still by “chip” bank cards. There are more efficient public-key techniques available these days, and these tend to be preferred for internet security, and are now starting to be deployed in bank cards.
I see people mentioning cryptography, but there is actually another use that nature came up with. Cicadas only come out every so many years (depending on which brood it is) and they do this to avoid running into each other and competing for resources. They can’t just coordinate one group on one set of years, and another on another set, so they each independently came out after a prime number of years to avoid competing with each other. 5 and 7 are common numbers of years, so they only come out at the same time every 35 years. There are also some that come out every 3 years, which prevents 9 years from being a cycle, because the 9 year cicadas would always have to compete with 3 year cicadas.
Prime numbers are a bit like the elements of the periodic table but for maths. All other integers are made by combining them via multiplication, so much so that these are known as *compound* numbers. So I suppose you could ask what the actual real world application of the periodic table is. If you’re not doing anything chemistry related, then it’s useless (except in pub quizzes). But if you *are* doing something to do with chemistry, then it’s absolutely critical and foundational as it explains a bunch of important relationships that are needed for the world as we know it to work. Primes are like this but for maths.
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