Why prime factorization ?

[ErnestoNet]

Interesting web resource: GIMPS search for the biggest prime: http://www.mersenne.org/[^] The latest maximum prime has 12,978,189 digits. Also, why look for prime numbers: http://primes.utm.edu/notes/faq/why.html[^]

it´s the journey, not the destination that matters

[Kenneth Haugland]

Quote:

Why this unique ability for prime numbers ?How is it possible that, any number can be expressed as product of prime factors ?

Both of these two questions could be answered by the fundamental theorem of aritmatic.

Quote:

What is it, which makes these prime numbers special ?

You could read my article, and there are lots of referances there. :) Finding prime numbers[^]

[Kenneth Haugland]

Wheter or not to include 1 in the list of prime numbers is debated among mathematicians. There are arguments to include it, and argument to not include it. 1 cant be written as a product of smaller primes except 1*1 However 1*N = N so you could always write any nyumber as a product of two primes if that was the case.

[Lost User]

Math is broken.

ihoecken wrote:

This is no equivalence, because there are no members and so both statements are correct: It contains no non-prime member AND it contains no prime members.

Why is that a problem? That's just the result of a vacuous truth. ∀x∈X:P(x) and ∀x∈X:¬P(x) can both be true, that just implies that X is the empty set. No problems there. And the equivalence ∀x∈X:P(x) = ¬∃x∈X:¬P(x) is a real thing. So there you go, the statements are equivalent, but that means that an empty set can be typed (because a type is just a predicate as well - the elements of the empty set are of all types simultaneously but lets not get hung up on that, they don't exist anyway), and thus math is broken. QED.

[Ingo]

harold aptroot wrote:

So there you go, the statements are equivalent, but that means that an empty set can be typed (because a type is just a predicate as well - the elements of the empty set are of all types simultaneously but lets not get hung up on that, they don't exist anyway), and thus math is broken. QED.

The proof for the equivalence is true, but the usage of equivalence for our problem is not. BobJovana said: 'the empty set contains only primes' is equivalent to 'the empty set has no non-prime members'. But:

harold aptroot wrote:

∀x∈X:P(x) = ¬∃x∈X:¬P(x)

That would be ∀x∈X:P(0) = ¬∃x∈X:¬P(0) Statement and proof are not corresponding!

------------------------------ Author of Primary ROleplaying SysTem How do I take my coffee? Black as midnight on a moonless night. War doesn't determine who's right. War determines who's left.

[Lost User]

So maybe math is not broken, that's always nice..

[Lost User]

I don't get any of these arguments. There is nothing in it, yes, so what? "The nothing" can, so far, still be "zero prime numbers". Or zero of anything else for that matter because the same vacuous truth of "all elements are of type t" is true for all t. The definition of the empty set doesn't say anything about that.

ihoecken wrote:

Mathematically it's not true that it contains prime numbers. The definition says: There is nothing in it.

It doesn't have to contain prime numbers, it only has to contain only prime numbers, which is the same as saying that all numbers in it are prime, which is vacuously true.

[Vijay Rajanna]

Thanks a lot, your article is very informative. :thumbsup:

Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]

[Ingo]

harold aptroot wrote:

It doesn't have to contain prime numbers, it only has to contain only prime numbers, which is the same as saying that all numbers in it are prime, which is vacuously true.

No! It contains nothing of everything. So it contains no prime number and no not prime numbers. And that is obviously true. :rolleyes: I think we won't come to an agreement. Let's say there are differences between us, but I could agree that there are nearly nothing ;)

------------------------------ Author of Primary ROleplaying SysTem How do I take my coffee? Black as midnight on a moonless night. War doesn't determine who's right. War determines who's left.

[BobJanova]

You posted links to the definition of 'empty set', which is not in question. I think you're having trouble with English. Let's introduce a bit more maths language into the sentence. 'Set S contains only X' is equivalent to 'Set S is a subset of the set X'; in this case X being the set of primes. The empty set is a subset of every other set, so 'The empty set contains only X' – equivalent to 'The empty set is a subset of X' – is true for any X.

[Member 2053006]

Vijay Sringeri wrote:

Why this unique ability for prime numbers ?

It is not a unique ability for prime numbers, it is just that these numbers can not be sub-divided any further. You can find the LCM and HCF using any numbers, but they will always be a combination of prime factorials, so using prime numbers is far easier.

Vijay Sringeri wrote:

How is it possible that, any number can be expressed as product of prime factors ?

Essentially because a prime number can not be divided and a non prime number can be. Any number n that is not prime has at least two divisors that are not 1 and n. These divisors are either prime or non prime. If they are non prime then by definition they follow the same rule as n. These numbers are smaller then n, so repeating this rule will always result in only prime divisors.

Vijay Sringeri wrote:

What is it, which makes these prime numbers special ?

The fact that they are prime and can not be divided.

[BillWoodruff]

The complete lack of any mention of Zero in this discussion has sucked out all meaning for me, and left me inside a total vacuum. Since Zero multiplied, or divided (except of course Zero divided by Zero), by any number, natural perverted, or even fractional, will always be Zero: therefore Zero is the Prime of Primes, not to mention that Zero raised to any power remains Zero, not to mention that subtracting Zero from, or adding Zero to, any number leaves the number unchanged ! That any number divided by Zero is an infinity (whose ordinality, or Aleph, among other possible infinities: is ultimate ?) which cannot be conceptualized within linearly digital Turing/Von Neumann theoretical computational design, and must be expressed by some "place-holder" like "undefined," or "NaN," or will, on a practical level, in many circumstances crash a computer: is proof of its sacred power. Zero is the unique singularity of the transition between positive and negative numbers, thus equivalent to the Omphalos, the stone of the navel of the geo-body of the cosmos, which for the ancient Greeks was located at the shrine of the oracle at Delphi. I propose to you that the infinite set of all possible prime numbers is contained within the infinity created by Zero divided by Zero like a tiny foot in a huge shoe: lots of wiggle-room no matter what #1 does, or does not, do. best, Bill

"Takuan Sōhō died in Edo (present-day Tokyo) in December of 1645. At the moment before his death, Takuan painted the Chinese character 'meng' ("dream"), laid down his brush and died."

[Sigismondo Boschi]

Because the "secret" factors A*B = C, with A prime and B prime, are the holy grail to break RSA, the asymmetric keys algorithm used world while to enforce security, with SSL, HTTPS, VPNs... Ruffly speaking, in RSA itself, C is "the public key" and its factorization, A and B "the private key". It is straightforward to find the factors for small numbers, but it happens that it is very hard to find such factors for large numbers (indeed - you need to extensively search for them). To date it has been possible to break up to a 768 bit RSA key (C is 768 bits long) by using a cluster of many hundred servers. Larger keys (1024, 2048 bits) are still considered secure (needed hundred years of cluster computing time for one single key) - and they will, until some strong improvement will be performed in number theory.

[Ingo]

BobJanova wrote:

You posted links to the definition of 'empty set', which is not in question.

I didn't started with empty set. It was used to support a statement, but it wasn't brought together correctly, so I stated the definition, which is proof that is doesn't support statement.

BobJanova wrote:

'The empty set is a subset of X' – is true for any X.

That I said. It's a subset. It's a subset of all primes. It's a subset of all non-primes. But it doesn't contain any non-primes nor does it contain any primes.

------------------------------ Author of Primary ROleplaying SysTem How do I take my coffee? Black as midnight on a moonless night. War doesn't determine who's right. War determines who's left.

[Lost User]

ihoecken wrote:

No! It contains nothing of everything. So it contains no prime number and no not prime numbers. And that is obviously true.

Yes I completely agree, I just don't see why this is an issue. This means it doesn't contain any non-primes, so it passes the test.

[YvesDaoust]

Mathematicians love to decompose things. The parts of a thing a usually easier to handle. When they considered the integers and the law of addition, they saw nothing interesting. Any integer larger than 2 decomposes into smaller parts: 12 decomposes into 1+11, 2+10, 3+9, 4+8, 5+7 and 6+6. This is trivial. 1 cannot be decomposed, so 1 is the only "prime" as regards addition. When they considered the integers and the law of multiplication, things got funnier: 12 decomposes into 2 x 6 and 3 x 4, 6 decomposes into 2 x 3 and 4 decomposes into 2 x 2. But 2 and 3 cannot be decomposed. So the idea of primes came quite naturally by discovering thoses numbers that cannot be decomposed. And they found much less regularity, meaning cool things to study. There is no real mystery in the decomposition: either a number cannot be decomposed, then it is called prime, or it can be decomposed, then it is called composite. You can apply the same reasoning to its parts, which are smaller. In the end, you always end up with prime factors, because decomposition is a finite process. (This is a case of infinite descent[^].) More mysterious is the fact that the decomposition is unique: whatever the way you apply the decomposition, you always end up with the same factors with the same multiplicity. Example: 12 = 2 x 6 = 2 x 2 x 3 or 12 = 3 x 4 = 3 x 2 x 2. But you easily understand that if a prime factor p appears r times in one decomposition of an integer n and s times in another (let r < s), we have a problem, as if we divide n by p^s, in the first case we will get a nonzero remainder and in the other case a zero remainder ! What makes prime numbers so attractive is that mathematicians have failed so far to find simple structures in their distribution, leading to the famous Rieman's hypothesis[^], considered a major unsolved problem in modern mathematics.

[Ingo]

harold aptroot wrote:

Yes I completely agree, I just don't see why this is an issue

Well, perhaps we talked past each other. I just diagreed with that statement:

harold aptroot wrote:

an empty set contains only prime numbers

Of course I agreed with:

harold aptroot wrote:

1 is the product of { }
p is the product of { p } for p is prime

So I said nothing against the original statement of yours. I think that it's correct, I was just agaist the statement the empty set would contain only prime numbers.

harold aptroot wrote:

This means it doesn't contain any non-primes, so it passes the test.

I totally agree to that.

------------------------------ Author of Primary ROleplaying SysTem How do I take my coffee? Black as midnight on a moonless night. War doesn't determine who's right. War determines who's left.

[Lost User]

Ok, now this is getting somewhere. What's the issue with "an empty set contains only prime numbers" exactly? I wrote that in part to be funny really, but to me that statement means the same as "for all x in the empty set, x is a prime number", and that would make it not only amusing but true.

[Ingo]

harold aptroot wrote:

or all x in the empty set, x is a prime number

Now it's starting over from the beginning. Stating that is pointless, there is no x in the empty set, because the empty set has no elements by definition. So you can't say for all x in the empty set, x is whatever. You can say the empty set contains no not-prime number (as it containt no prime numbers), that of course is true. Nothing is not an element of anything. It's just nothing, by definition.

------------------------------ Author of Primary ROleplaying SysTem How do I take my coffee? Black as midnight on a moonless night. War doesn't determine who's right. War determines who's left.

[Lost User]

Of course you can say it, it's just a vacuous truth, and exactly equivalent to saying "there is no element in the empty set that isn't a prime number", no trouble there :confused: It's just the second case described here: http://en.wikipedia.org/wiki/Vacuous_truth[^] What you can't say is "there exists an element x in the empty set such that P(x) is true"