Question

Fundamental Theorem of Arithmetic::
```
Every positive integer can be
written as a product of primes (possibly with repetition) and any such expression is
unique up to a permutation of the prime factors. (1 is the empty product, similar to 0
being the empty sum.)
```
what is the meaning of "unique up to a permutation of the prime factors" and how do you prove it?

Answer 1

MATH 25 CLASS 5 NOTES, SEP 30 2011
Contents
1. Prime numbers and prime factorization 1
Quick links to denitions/theorems
The Fundamental Theorem of Arithmetic
1. Prime numbers and prime factorization
We've spent a good amount of time discussing divisibility, solving ax + by = c in
integers, and greatest common divisors. We'll now put what we've learned to good
use by studying properties of prime numbers more closely.
It all starts with Euclid's Lemma, which I'll repeat again:
Lemma 1 (Euclid's Lemma, 2.1b, 2.2). Let p be a prime. If pjab, then pja or pjb.
More generally, if pj(a1 : : : an), then pjai for some i.
Proof. We saw the proof for n = 2 last class. The proof for general n uses induc-
tion. Suppose we know this statement for a given n. We want to prove the corre-
sponding statement for n + 1 terms; that is, we want to prove the statement that if
pj(a1 : : : an+1), then pjai for some i. Let a = a1 : : : an; b = an+1. Then pjab =) pja
or pjb. If pjan+1, we are done. If pja = (a1 : : : an), use the inductive hypothesis to
conclude that pjai for some 1 i n.
A quick corollary of Euclid's Lemma is the following:
Corollary 1 (Exercise 2.1). If p is a prime, and pjak, then pja.
This is easily proven by just applying Euclid's Lemma to a1 = a; : : : ; ak = a.
Why is Euclid's Lemma so important? It allows us to prove the following theorem,
one of the most important in the class:
Theorem 1 (The Fundamental Theorem of Arithmetic, Theorem 2.3). Let n > 1 be
an integer. Then there exists a unique prime power factorization
n = pe1
1 : : : pek
k ;
where p1; : : : ; pk are distinct prime numbers, and ei > 0 are positive integers. (When
we say this factorization is unique, we really mean unique up to permutation of the
prime-power factors. For instance, we consider 12 = 22 31 to be the same factoriza-
tion as 12 = 31 22, since we just moved around the powers of 2; 3.)

Answer 2

looks like copy and paste ... i was looking for discussion.

Answer 3

there is no copy and paste that was a writing piece my brother wrote

Answer 4

you mean you copied your brother's note lol

Answer 5

well yup because he wrote it on my computer so that's wat happens lol

Answer 6

I am aware of the theorem and proof ... there is a term "unique up to a permutation of the prime factors" I cannot decipher what it is trying to tell.

Answer 7

It possibly means this:

The prime factorization of \(N\) is unique and we consider \(N = a^x \cdot b^y \cdot c^z = b^y \cdot a^x \cdot c^z = c^z \cdot a^x \cdot b^y\), i.e., all the --permutations-- of writing the product are considered to be the same. They do not destroy the uniqueness of the prime factorization of \(N\).

Answer 8

prime factorization of 6 can be : 2*3 or 3*2

Answer 9

the prime factorization is not unique if order is considered

Answer 10

why not just write "permutation of prime factorization is unique" why write the term "up to"??

Answer 11

English is a funny language.

Answer 12

tru that ^

Answer 13

ahh i saw that phrase "upto" in many places... let me see if i can recollect one more example of using "upto"...

Answer 14

haha ... very funny

Answer 15

should have posted in English section ...

Answer 16

I think it means something along the lines of

"up to considering the fact that different permutations are different things, they're not considered to be unique... but even after all that, they *are* unique."

Answer 17

here :
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/18_022007L07.pdf


two different examples... search `up to` :)

Answer 18

I is very very confused. Engliesh is such a wired language ... lol

Answer 19

"any such expression is unique up to a permutation of the prime factors"

This means that they're defining "unique" up to a point that different permutations of writing the prime factorization will be the same.

Meaning that the expression \(a^n \times b^m\) and \(b^m \times a^n\) are considered to be the same.

Answer 20

hold on a sec ... lemme decipher the paper ganishe posted.

Answer 21

```
That is the solution. And
any multiple of that is a solution. If you like to neatly simplify them you could say
negative one, one, negative two. If you like larger numbers you can multiply that by
a million. That is also a solution. Any questions about that? Yes? That is correct. If
you pick these two guys instead, you will get the same solution. Well, up to a
multiple. It could be if you do the cross-product of these two guys you actually get
something that is a multiple
```

Answer 22

is that transcript of Dennis Aurox @ganeshie8

Answer 23

Yes ! he uses `up to` a lot lol, atleast one time in each lecture :P

Answer 24

lol, I lost it at "the cross-product of these two guys"

Answer 25

I cannot trust that guy ... I speak better English than him lol. he is heavily french accented.
JK

Answer 26

yes lol :) he even has a translator for composing his class lectures

Answer 27

no american online??

Answer 28

haha ... if he probably says then it must be correct ... after all he is professor at MIT.

Answer 29

i just love that guy lol,
i think he was using `up to` word in the context of saying something like below :

taking cross product of any two vectors lying on a same plane gives the same normal vector up to scaling...

Answer 30

looks like someone posted the same question here.
http://math.stackexchange.com/questions/473420/what-is-the-difference-between-being-unique-unique-up-to-isomorphism-and-unique

Answer 31

lol ... these days, even you can't be the first stupid on internet. I find my problem solved even before I post. the first sentence of second paragraph solves my question. who wants the medal?

Answer 32

let's make 2x2 for both of you!!

Answer 33

hahaha that hilarious you cant be the first stupid on internet xD give it to @ikatouni expX

Answer 34

ikatouni = best answer ever.

Answer 35

well well ... maybe one of you help him :) i hate to unto.

Answer 36

exactly were is my medal