system
Given the following post, respond as if you're commenting on Reddit:
human
What is the importance of the prime numbers?

I've often wondered what the importance of prime numbers are, since they seem to be stressed in mathematics. I've asked this question many times (this is the first time I've asked reddit), and I either get my question "answered" with a question, or a smarmy answer.  

Can someone please explain to me the applications and potential benefits from, say, one day finding *the* prime number formula? I do know that there are applications for this in cryptology, but outside of this, I'm not sure what the implications are or what the importance is. Thanks in advance for any help! This question has always nagged at me.
gpt
Just to give an illustrative example, one of the most important properties of the natural numbers is the Fundamental Theorem of Arithmetic, which says that every natural number has a unique prime factorization. This fact was crucial to Gödel's proof of the Incompleteness Theorems, which is widely considered to be one of the deepest results about mathematics that has ever been proved. Gödel's Incompleteness Theorems state that any axiomatic formal system that is powerful enough to capture arithmetic will always be either incomplete (in the sense of not being able to prove everything that's true) or inconsistent (in the sense that one can prove a contradiction), and that no such system can prove its own consistency. Gödel's famous proof involves attaching a number to each symbol in the formal language, and then storing these numbers as the exponents on an increasing sequence of prime numbers. This "arithmetization" assigns a number to each statement in the formal system, and because of the Fundamental Theorem of Arithmetic such an assignment is guaranteed to be unique. Given a number, we can write the statement that it represents, and given a statement, we can calculate the number that gets assigned to it. Arithmetization gives the formal system a *recursive vocabulary*, meaning that the system can say things about itself. For example, Gödel proved that there is a statement which says of itself "This statement is not provable." If the statement is false, then the system is inconsistent, and if the statement is true, then the system is incomplete. Gödel proved that this is an inherent property of such systems, and that there is no way of getting around incompleteness (such as by adding new axioms). This groundbreaking and frankly shocking result of metamathematics depends entirely on the properties of prime numbers.