We had to memorize them in school, but I never knew why. I know what they are (not divisible by another number) but don’t know why they are so important and studied.
We had to memorize them in school, but I never knew why. I know what they are (not divisible by another number) but don’t know why they are so important and studied.
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They are used in Cryptography for one example. They are used to generate key pairs for transmitting data safely across the internet via public/private key encryption.
Odd that you had to memorize them. Are you sure you weren’t just expected to be able to recognize one, or determine if a given number was prime? I can’t think of a single reason you’d need to memorize them for school.
Every single number greater than 1 is either a prime number number or a composite number. This means with only the prime numbers, and 1, you can make every other number through multiplication.
Studying and proving assertions about the nature of prime numbers – including that they’re infinitely many primes, their patterns, and proving primality – have been useful for advancing mathematics and our understanding of number space.
Say you have 15 people over and you want to break them up into groups. You might do 3 groups of 5. If you have 16 people, you might do 4 groups of 4. But what about 17?
Divisibility is something that just tends to come up a lot, and prime numbers are the “inverse” of that. All the numbers that aren’t on your times tables.
You could talk about specific use cases regarding certain algorithms, cryptography etc. Which is all fair. But really, as humans we just like dividing stuff up and find it interesting when you can’t do that.
They are important because they are non divisible. You got that part.
Kind of the more important way to phrase this is the vast majority of other numbers (composite numbers) are “made up” of prime numbers.
Like 1961. Do You know if that number is made up of two other numbers or is it prime? can you tell me? Can you figure it out without guessing exhaustively?
You really can’t. That’s a huge deal. We can’t come up with a way to figure out if a number is prime without brute forcing with known composite numbers.
1961 is 53 * 37.
If you do this with huge random prime numbers it’s next to impossible for someone, even with a computer to figure out which two you started with.
This is one of the keys of some computer cryptography.
Also the primes KINDA follow a pattern. People have made many visualizations. It’s interesting because it’s not even visualizing physical phenomena, just pure logic, and it looks like there’s a “shape” to it.
They are fascinating things and they show up in nature – esp biology.
How do you memorise an infinite number of numbers?
The study of prime numbers is an incredibly rich field. The main reason primes are important is the appropriately named Fundamental Theorem of Arithmetic. That every number has a *unique* prime decomposition.
So 12 = 2* 2 * 3. 2 * 2 * 3 is only ever 12, and 12 is only ever 2 * 2 * 3. It may seem trivial, however if we don’t restrict it to primes, then 12 = 1 * 12 = 2 * 6 = 3 * 4 = 2 * 2 * 3. So there is no unique way to represent 12 as the product of numbers, but there is for primes. You can then see how the ancient Greek, Euclid, proved it for all numbers, https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic#Proof
You can use this to do things like map two lots of the number line onto the number line. By mapping (n,m) to 2^n 3^m . As prime factorisation is unique, there is no repetition, so you’ve mapped it on to the number line. This wouldn’t work if we used 2^n 4^m, as (2,0) maps to the same point as (0,1), as 4 is not prime.
There are other things, like if you consider doing maths on only a prime number of numbers, like a 13 hour clock. Where 10 + 4 = 1, and 4 * 4 = 3 and such, you can guarantee that there is always another number that you can multiply (other than with 0) with to get 1. So for 5 numbers: 1 * 1 = 1, 2 * 3 = 1. 3 * 2 = 1, 4 * 4 = 1. This doesnt work if you have a non prime system.
It’s pretty difficult to explain without going into a lot of pretty complex math, but because prime numbers can only be divided by themselves and numbers having no common factors is extremely important in a huge variety of math prime numbers are extremely important.