I've recently read about the Fundamental Theorem of Arithmetic and I think that I have just about understood the proof. What I found quite interesting at first was the "Fundamental" part in the name. I understand that my mathematical experience is nowhere near enough to fully appreciate this theorem, but can someone can explain to me why this theorem is so important? What would happen if this theorem wasn't actually true?
Thanks in advance!
On
The fundamental theorem of arithmetic is important because it tells us something important and not immediately obvious about $\mathbb{Z}$ (the ring of the counting numbers together with those numbers multiplied by 0 or $-1$). Every nonzero number in $\mathbb{Z}$ can be uniquely factorized into primes without regard for order or multiplication by units.
It doesn't matter if you consider numbers like $-2$, $-3$, $-5$, $-7$, etc., to be prime or not. For example, $-2 = -1 \times 2 = 2 \times -1$, and we see that 2 is a positive prime and $-1$ is a unit, and it doesn't matter which one we present first. Accepting some negative numbers (specifically the positive primes multiplied by $-1$) as primes presents no philosophical problem to the fundamental theorem of arithmetic.
This is something that you must not take for granted when you step out to the larger world of algebraic integers. Consider for example $\mathbb{Z}[\sqrt{-5}]$, which consists of numbers of the form $a + b\sqrt{-5}$, where $a$ and $b$ are integers of the kind you're already familiar with. For now, take my word for it that $$21 = 3 \times 7 = (4 - \sqrt{-5})(4 + \sqrt{-5}) = (1 - 2\sqrt{-5})(1 + 2\sqrt{-5}).$$
Well, you can verify the equation with Wolfram Alpha, for example, put in (4 - Sqrt[-5])(4 + Sqrt[-5]). But for now you still have to take my word that the equation above does not contain any units, nor any non-obvious multiples of units.
This is the sort of thing that would happen in $\mathbb{Z}$ if the fundamental theorem of arithmetic was not actually true.
To help us make sense of integral domains like $\mathbb{Z}[\sqrt{-5}]$, we use a "norm" function that takes in a number from that domain and gives us a number in $\mathbb{Z}$. The norm function for $\mathbb{Z}[\sqrt{-5}]$ is $N(a + b\sqrt{-5}) = a^2 + 5b^2$. Thanks to the norm function, we can see that, for example, $1681 = (31 - 12\sqrt{-5})(31 + 12\sqrt{-5})$ is an incomplete factorization, because $(6 + \sqrt{-5})^2 = (31 + 12\sqrt{-5})$ (we just had to find a "smaller" number with a norm of 41).
The fact that $\mathbb{Z}$ has unique factorization (as shown by the fundamental theorem of arithmetic) allows us to prove things about numbers in other integral domains regardless of whether or not those other domains have unique factorization. And it turns out that of the infinitely many imaginary quadratic integer rings, only nine of them have unique factorization (clearly $\mathbb{Z}[\sqrt{-5}]$ is not one of them).
Let us also take a look at the frivolous theorem of arithmetic and the fundamental theorem of algebra.
The frivolous theorem of arithmetic tells us that almost all positive integers are very large. This is obvious, and therefore frivolous, because for any large positive integer that you can name, I can name a larger still positive integer just by adding 1, e.g., you say "googolplex", I say "googolplex plus 1".
The fundamental theorem of algebra tells us that every valid polynomial has as many roots as its degree. For example, $x^4 - 1$ is a polynomial of degree 4, and it has four roots, two of which you should be able to figure out yourself. The fundamental theorem of algebra seems like it should be obvious, yet its proof is quite lengthy and involved. But to appreciate its meaning and importance, you just need to understand imaginary and complex numbers.
On
I decided to compare this theorem to another favorite theorem of mine, Fermat's little theorem (which some people just call "Fermat's theorem).
On ProofWiki, I went to the pages for both theorems and clicked "what links here": https://proofwiki.org/wiki/Special:WhatLinksHere/Fundamental_Theorem_of_Arithmetic , https://proofwiki.org/wiki/Special:WhatLinksHere/Fermat%27s_Little_Theorem
As you can see, the fundamental theorem of arithmetic is invoked for the following results:
Compare Fermat's little theorem:
I have chosen to ignore lemmas, corollaries, definitions, biographies, and of course pages that are very specific to ProofWiki, like the Help:Page Editing page.
My conclusion is, that by this measure, Fermat's little theorem is important in the sense that other theorems depend on it, but not as important as the fundamental theorem of arithmetic.
For me, it's important because it tells you what numbers are made out of! It tells you that prime numbers are the "building blocks" of every number, and even better, this prime factorization is unique. So this tells you that a number just "is" a product of primes.
Aside from being pretty cool (I think the proof is neat and easy to understand for how important it is), think about how useful this was early on when you were trying to add together fractions; you find the least common denominator usually by using the prime factorization of the denominators. Similarly, when you want to reduce the ratio of two integers, it can be useful to divide both the numerator and denominator by the greatest common factor, which again can be found using the greatest common factor. If it weren't true you would have to find another algorithm! In other areas I know very little about, like cryptography (by the way, fun to read about when you are learning elementary number theory), the fundamental theorem of arithmetic gives you a good way to send secret messages to another person just using knowledge of one of the prime factors of a really big number. We always know this prime factorization exists, but it can be very hard to actually find it for a really big number.
As people are hinting at in the comments, this idea generalizes to other areas in abstract algebra (a part of math where you study sets of things with different levels of structure on them); it can be useful to try and find the building blocks of other sets. In fact building other (finite) fields, which are sets of numbers that are a lot like the rational and real numbers in their level of structure, relies on using building blocks of polynomials and a euclidean division similar to the one you use when you divide integers by other integers. When I started learning about that stuff, I found it really helpful to think back on how these processes worked for things I found more familiar, like integers.
I haven't gotten very far in math, but I have found that concepts like "how do we reduce this one thing that we don't know much about into just a collection of smaller things we do know something about" come up a lot across subjects. It helps to have seen them before and have a good stock of simpler examples in easier to grasp or more familiar places.