When talking about the theorem Gowers here wrote that the factorization of 36 is 2 x 2 x 3 x 3. But the composite number 36 by that notation looks like a product of a number and a composite number (2 x 18), a product of a composite number and a composite number (4 x 9), or a product of a composite number and a prime number (12 x 3). A notation like x(2, 2, 3, 3) or (2, 2, 3, 3)x doesn’t get used all that often when talking about the Fundamental Theorem of Arithmetic. And thus, it’s not so obvious, because authors don’t use appropriate notation to theorem all that often when talking about it or make it clear that the theorem implies that for any composite number, there exists of a k-ary product of prime numbers only, where k is some natural number.

]]>1) a is prime means there exists no number between 2 and a-1 that divides it

2) there exists a prime factorization of every integer (algo exists for calculating it)

3) let a prime factorization be pi0 * pi1 * … pin

let a different prime factorization be pj0 * pj1 … pjm

pi0 * … pin = pj0 * … pjm

pi0 must divide pj0 * … pjm

thus pi0 must divide some pj, since it itself cannot be written as a product of 2 or more primes

this is a contradiction, so there cannot be a different pj0 … pjm

For other rings however this makes perfect sense. Consider for example the ring of polynomials. What can you divide x + 3 by? ]]>

So why use imaginary numbers in your argument at all?

]]>I looked at https://en.wikipedia.org/wiki/Modulus_(algebraic_number_theory) but like many other wikipedia articles on math it was not very helpful. ]]>

Let’s suppose that there are numbers that have more than one factorisation. Then there must be a smallest one. Let’s say it’s C. Then C = A = B, where A = a_1*a_2* … *a_N, and B = b_1*b_2*…*b_N, and all the a’s and all the b’s are prime.

Every a is different from every b. If that were not the case then there would be a common factor, which we could divide by, to get a smaller C with more than one factorisation. But C is the smallest one.

A is divisible by a_1. Therefore B must be divisible by a_1. Therefore a_1 must divide one of the b’s, b_i (say). Then b_i is not prime. But all the b’s are prime. This is a contradiction. So our supposition that there are numbers that have more than one factorisation is incorrect. QED.

]]>was good. I do not know who you are but certainly you are going to

a famous blogger if you aren’t already 😉 Cheers! ]]>

Proof: a:b=c:d, then

ac:ab=c:d, then

ac:bc=ac:ad, so

bc=ad.

Just one possibility of non unique prime factorization, but with the conclusion accepted, indicating the nonexistence of primes with the ratios. ]]>

We should always think about a number as a cardinality of a power set. This way we can understand better why 2^0=1. From the fact that cardinality of an empty set is 0, we can say that the power of this set is 1 since it contains only one element, the empty set. And, then, for all integer n, n^0=1 …

]]>Convincing people that obvious-sounding things aren’t really obvious, but are still true, is a psychologically difficult chore. “It isn’t obvious, but it’s true, and if you follow this incredibly complicated line of reasoning you’ll see why” – that doesn’t sell, because it seems you wind up just where you started (knowing that something is true) after having done a lot of extra work. There are not many situations in life where it’s worthwhile going to elaborate pains to verify something you’re already sure is true.

This sort of thing appeals to pathologically careful people, and these people may be born mathematicians. There are other people who can learn the appeal of being pathologically careful, as a kind of mental discipline or game, and maybe these people are the target audience for a course like this. There are other people who can understand the idea of being pathologically careful, but find it a tiresome chore. And there are others who don’t get it all. For the latter two types, I try to make sure my course has a lot of other fun stuff in it!

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What’s the pattern here? Each number is twice as big as the one before it. So what should the first number be? It should be 1, since that will continue the pattern. Having it be 0 would ruin the pattern.

If you keep the pattern going, will work just like the other powers of 2 and you life will be happy. If you destroy the pattern by decreeing that , this power of 2 will act different from all the rest, and your life will be forever miserable (until you quit doing math, but please don’t do that).

One important thing to realize is that in math, we get to make up the rules. It’s *not* true that God decreed and it’s our job as humble servants to memorize this rule. We make up the rules. But we try very hard to choose rules that give simple patterns. ‘Exceptions’ are not nice.