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.
In: 20
Lets say you want to make a gear, You have a dividing head with a 40:1 gear reduction(40 turns of the knob = 1 turn of the workpiece), and there are plates that let you divide one turn of the knob by the number of holes around a circle. If you want to make a 40 tooth gear, you turn the knob 1 full rotation between each gear tooth you cut.
Lets say you’re making a clock that lists what day of the week it is, and need a 168 tooth gear. How much do you turn the crank between each cut?
Prime factorization is useful here, 168 ends in an even number, so it’s divisible by two. 84 is still divisible by 2, 42 is still divisible by 2, 21 is 3 * 7.
So we need 2 * 2 * 2 * 3 * 7 divisions.
we have 2 * 2 * 2 * 5 from the 40:1 gear reduction. We’re missing 3 and 7, so we need a plate with a 21 hole pattern in it(or some multiple thereof). Now we have that leftover 5, so we only use every fifth hole in the plate to rotate the gear the correct amount for the next cut.
The rich theory we have about prime numbers is useful for systems we can model with integers, at which point those abstract statements become real statements about the system.
A lot of our technology has been intentionally made discrete by us because it is easier to think about, leading to the various applications here (cryptography, error correction, hash tables, data encoding).
A lot of the natural world is continuous, so it can’t be modeled very well with integers. However, many things do end up approximating discrete processes. Years and seasons give biological systems a discrete structure to hang onto, so you end up with the situation someone mentioned about cicadas, who use the divisibility properties of prime numbers to get a biological advantage. Small quantum systems also form discrete structures, so you occasionally get [situations where a physical structure reflects some fact about primes](https://physics.aps.org/articles/v5/s63).
In terms of all the times humanity uses prime numbers, though, by far most of it is in doing other forms of discrete math. We teach them in school because we as a society have decided it’s valuable for people to have some background in math, even if they don’t end up becoming mathematicians, and primes follow very closely when you have learned how to count and do basic arithmetic.
An alien civilization could send signals which repeat every increasing prime number to signal their presence.
Yes, so sad, that you mistyped a comment that didn’t track logic and then when addressed on your mistake you continue to assert that your right. Very sad.
But good news! You are right! Congratulation, Internet winner! You’re superior and right when when your typing is wrong!
To say they have uses in online encryption is a big understatement.
Prime factorization is the entire basis of internet security.
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