Who cares about prime numbers? Like, how are they useful?

That's not an argument in favor of prime numbers, nor a particularly remarkable statement.

3*3?

Scheme Theoretic geometry has resulted in many remarkable statements, and it is all done over SpecZ.

true. what about 12? hows that a product of primes?

3*2*2

Okay. So all you're saying is that every ring is an abelian group. Wow, I sure care about prime numbers now!

That is not at all what I am saying. The points of SpecZ are 0 and all prime numbers.

that's cheating. it should only be limited to a set of 2 prime numbers.

every number is composed of primes

they make up the universe

Holy shit, congrats on learning lots of definitions. All affine schemes have primes ideals as points, and the definition of a prime ideal generalizes the notion of prime integers. The reason we do that is because we think of the set underlying a scheme as the set of irreducible subvarieities. That Spec Z is final in schemes just says that rings are Z-algebras, so basically just that they're abelian groups with multiplication
You're not saying anything noteworthy.

>All affine schemes have primes ideals as points

Yes that is the definition

>That Spec Z is final in schemes just says that rings are Z-algebras

Yes it an equivalent statement, but that doesn't make the idea that every geometric space has a natural morphism to the set of prime numbers any less interesting.

Yes it does. The fact relies on us generalizing the notion of prime integer in the first place, so that comes from some inherent interest. It's just such an ass backwards reason to give for being interested in prime numbers.

>it should only be limited to a set of 2 prime numbers.
Lol what? You're saying this based on what? Because you're wrong, it's definitely not.

I know it is backwards, but by itself the idea of geometry having a fundamental connection to primes is interesting. It not a priori intuitive why a set of prime ideals would be of any interest geometrically.

just makes my point. if the set isn't limited to an actual limited number; then the logic is basically:
>dude, numbers are made out of other numbers

shut the fuck up you imbecile, you have no point. every number is a finite product of primes, yes it's literally "numbers are made out of prime numbers" and prime numbers are simpler in very strong ways.

you're not smart enough to see the uses that are right in front of you. that's all. dumb people do that. this is literally the answer.

Number Theory is interesting, but maybe not directly useful. But there is something unique to number theory:

Number theory generates the hardest problems. All the hardest problems have something to do with integers, and they are easy to state which gives those types of problems some kind of intrinsic appeal.

Think of Fermat's Last Theorem. The only one who could prove it through elementary means was himself, but because there was not enough room in the margin we had to come up with another proof. What did it take? Hundreds and hundreds of years of algebra. If you wanted to learn all of algebra today you'd take decades. If you wanted to learn all of algebra back in Fermat's time, you could do it in an afternoon.

Number theory changes the earth in the most beautiful ways. Even today, the hardest problems still come from number theory. Analysts can jerk themselves off with derivatives all day for all I care, but until they prove something related to a number theoretic function they are all brainlets in my eyes.

With prime numbers you can do all kinds of amazing things. As for example, this post number is a prime.

Fuck, almost.

>even

Some didn't graduate elementary school. Let me show you a real prime number, like this post number.

Fermat almost certainly did not prove FLT. We believe he had a flawed proof, and we believe we know what it would have been. It's basically the fact that factorization is weird when you enlarge the integers.

I did it

I was fucking joking man. Come on.

That wasn't a very funny joke.

No, no. This is a prime number. Check em.

What set? You are making up definitions as you go along, you have no idea what you are talking about.

primes checked

1 is not a prime

8823317 and 8823301 are, you dumbfuck

You can probably use them as generators of complexity

>hurr durr what is RSA?

Are we being raided or something?

...

OP here. I'm just playing around lol. Had no idea so many would bother

As soon as someone invents a formula for factoring primes, you're fucked.

They should focus on unbreakable cryptography rather than this math circlejerk.

You're just autistic

How do we know there are infinite primes? Why isn't it possible for there to be no more primes past some arbitrarily large number?

How about safely encrypting your bank information. Take a discrete math class.

Cryptography

Reductio.
math.utah.edu/~pa/math/q2.html

>writing this shitpost from a secured computer
>not understanding the value of primes

damn son

So it gives some other sources, but the source itself seems to contradict itself, by giving a rule and then showing how it is broken immediately afterwards?

>formula for factoring enormous primes
yea probably not

>unbreakable cryptography
>not math

just stop posting. what do you think cryptography is based on?

BTFO'D

holy shit.

Give me prime factorisation of 1 and 0

And then, is 3 a product of primes?

3^1

The reason primes are significant is that they can't be broken up into factors. (the number times 1 doesn't count as breaking it up)

1 is not prime though

And why do we care for factors?

at first i thought OP was joking, but now I know he is genuinely retarded

Literally: KYS

This is true. Three, however, is prime.

It's kind of curious that there's always a larger prime. One might think when natural numbers get large enough they're bound to be composite. This isn't the case.

>yea probably not

Isn't this a research area?

Also, yea is an archaic word for yes. You mean "yeah"

>how are they useful

Not useful in the slightest. You need to remember that math and numbers are really terribly representations of the real world. It is pretty much a religion for anything beyond applied maths.

This is what people in every discipline in the world do. Instead of diving into understanding an automobile, work on first understanding the simplest pieces comprising it.

>If you take two [..] primes, and multiply them together, you get a [..] number with only two factors.
Proof?

>by giving a rule and then showing how it is broken immediately afterwards?
yeah that confused me too. someone explain how this """proof""" is valid?
i mean, you could say that in this case, there actually was a bigger prime found, namely 59 (which is bigger than 2,3,5,7,11 and 13). But that's not how the proof works.

wwwf.imperial.ac.uk/~buzzard/maths/research/notes/inductive_proof_of_fundamental_theorem_of_arithmetic.pdf

I think if you reject the FTA you must reject Euclid's proof

I was reading that and thought you misunderstood my question, because I wasn't asking for a proof that "every number has a unique prime factorization", but that the product of two primes only has two factors.
I see now that you are implying that these are basically the same statements, because say 7*5=35 will only ever have the prime factors 5 and 7.
But what if i multiply 2 and some other prime number, and then it just so happens that the number is divisible not only by these two prime numbers, but by some other number as well?
Say you have 2*2*7 = 28. Then that's divisible by 4, and by 14 as well. Just as an example.

then it has two different prime factorizations

>Use "yeah" not "yea"
Jesus! Nitpick much? They both denote affirmation.

The point is that if there were only finitely many primes, then the fundamental theorem of arithmetic is false. But the fundamental theorem of arithmetic is very much not false, so QED.

would it be possible for multiple sets of prime numbers to have the same product?

For example

Prime Number A * Prime Number B = N
Prime Number C * Prime Number D = N

If you're asking what I think you're asking, no. The set of primes, with multiplicity, in a prime factorization it's totally unique for every number.

Every composite natural number has a unique(i.e. Different than any other natural number) prime factorization. Every prime has only one prime factorization. 14 isn't a prime factor of 28.

Decidability used to be a research area in mathematics and look how that turned out.

Just because people are doing research in an area doesn't mean that they will find anything. Some prime numbers appear in semi-regular groups but there are always outliers.

>14 isn't a prime factor of 28.
That may be so. But the original statement was
>If you take two huge primes, and multiply them together, you get a very large number with only two factors. You could use the number as a public key, and use the factors as a private key, since finding the factors of such huge numbers is beyond impractical.
Who's to say it can't have more factors, but they simply aren't prime?

>Who's to say it can't have more factors, but they simply aren't prime?
your example of 2*2*7 was a product of 3 primes, that's why it can be split into 4*7 or 2*14. This situation is different, because they explicitly stated they are multiplying 2 primes. Think about it, if the number did have other non-prime factors, those non-prime factors would themselves be factorable into primes. This implies non-unique prime factorizations, which was proved false earlier in the thread.

Oh, you can factorize numbers in several different ways. But the whole point of the fundamental theorem of arithmetic is that you can only factorize certain numbers in a certain such-and-such a way. Specifically, if you want to break natural numbers down into prime numbers, for each natural number, there is exactly one, and only one, way of doing this.

Let's take a brainlet example, an object lesson: the number 91 "looks like it might be prime", but really has the unique prime factorization of 7 x 13. 91 is an example of a number whose prime factorization consists of exactly two distinct (not equal to each other) primes.

Now, you could say things like 91 = 0.5 x 182, or 91 = 1 x 91, or 91 = i^4 x 91. These are other ways of breaking 91 down into factors. But notice how in each case, the original "they all gotta be prime" requirement is by definition broken. these other expressions for 91 simply have nothing to do with the pertinent situation, and a quite natural one for humans to contemplate: where you break a positive whole number down into positive whole number factors. This is what leads inevitably, and in every case, not only to prime factors, but to unique assignments of such to each natural number.

Man, I really like your explanation, I can understand more now. You definitely know your stuff. Could you also explain how exactly the public can be used to encrypt something, and how the factors can be used to decrypt it back?
Thanks a lot in advance if you do this.

i dont think you know what im asking, i just meant if there are multiple pairs of prime numbers that can be multiplied to the same number, i assume it woud be a large number

in my example N would not be a prime and surely it would have lots of prime factors

You can't have two different pairs of primes multiplying to the same number though.

Guys, I have a formula for factoring arbitrary primes: let p be a prime. Then p's prime factorisation is given by p.

Pls don't steal

So the proof that there are infinitely many primes, basically goes like this, if i say it in a brainlet way:

If there were a finite amount of primes, then you can multiply them all together and get the number p as a result. p+1 is then either a prime or a composite number. if it's a prime, then the assumption is wrong. and if it's a composisite number, it must have a unique prime factorization, but none of the existing primes can be factors (because of the +1), therefore the assumption is still wrong.

correct?

>Who's to say it can't have more factors, but they simply aren't prime?
The Fundamental theorem of arithmetic states that a number with two prime factors has only those factors.

Huh, I guess I didn't think about that one

This is correct

I apologize for my pedantry

That's technically correct but you may be in danger of circular reasoning by invoking unique prime factorization to prove the infinitude of the set of primes.
I say "may" because whether or not there is circularity depends on your definition of the 'set of prime numbers' -- if you define it to be the set of numbers divisible only by 1 and themselves, then you need the infinitude of primes to prove unique prime factorization.

The airtight (and constructive) way to state the result, which is how Euclid originally did it, is to say that for any finite set of prime numbers there is another prime not in the set. The following link discusses this in more detail:

math.stackexchange.com/questions/842187/proof-of-infinitely-many-primes-clarification

So, three times what equals 3? 3p=3 holds only for p=1 which is not prime

But how is factorisation giving us any insight into how numbers work? We know 242 is 11*11*2, but so what? Numbers have some use on their own, and prime factorisation is useful only to autistic number theoreticians

A natural number is either prime or the multiple of 2 or more primes.

0 and 1 are neither prime nor product of 2 or more primes

Ok. Sorry. Zero isn't a natural number so your point on that is irrelevant. Primes/composites are generally understood to be features of natural numbers > 1