Why prime factorization ?
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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.
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.
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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.
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BobJanova wrote:
The empty set contains 'only prime numbers' in
No, that's wrong: http://en.wikipedia.org/wiki/Empty_set[^] Quote: "the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero" http://www.proofwiki.org/wiki/Definition:Empty_Set[^] Mathematically it's not true that it contains prime numbers. The definition says: There is nothing in it. Take a look of "Axiom of empty set" it states that there is only one empty set, no matter what you want to describe. If you have a set of colours {blue, red, green}, it's the same empty set. There is only one. Containing nothing. http://en.wikipedia.org/wiki/Axiom_of_empty_set[^]
BobJanova wrote:
I think it is, then Harold's statement is true.
It's wrong. As it's not the definition of the empty set. Read it, then you see.
------------------------------ 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.
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.
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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[^]
Thanks a lot, your article is very informative. :thumbsup:
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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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.
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.
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BobJanova wrote:
in this case 'the empty set contains only primes' is equivalent to 'the empty set has no non-prime members'
That is logically and mathematically incorrect. 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. Mathematically it's wrong, you can't change it. It has nothing to do with your interpretation: empty is empty. Nothing in there. If you don't believe ask another one who studied mathematics or your professor from university, they will say the same. Edit: By the way. If you can proove, that you are right. Do it. I will make my mind up, if you can. I gave you links to the definitions that support what I said. Do the same for a real discussion.
------------------------------ 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.
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.
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Hi, This is basically a math question, but very much applicable to many of the computer algorithms. I know the fact that, - Any integer can be expressed as product of prime factors. - Prime factors can be used to find LCM and HCF - Prime factors can be used to check whether a number divides "N" ( N - Integer ) - Etc.. Etc... My question is.. Why this unique ability for prime numbers ? How is it possible that, any number can be expressed as product of prime factors ? What is it, which makes these prime numbers special ? I just found this article on web, which was informative, but was little hard to understand. Can someone explain me this mystery behind prime numbers, any links web resources is much appreciated.
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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.
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Hi, This is basically a math question, but very much applicable to many of the computer algorithms. I know the fact that, - Any integer can be expressed as product of prime factors. - Prime factors can be used to find LCM and HCF - Prime factors can be used to check whether a number divides "N" ( N - Integer ) - Etc.. Etc... My question is.. Why this unique ability for prime numbers ? How is it possible that, any number can be expressed as product of prime factors ? What is it, which makes these prime numbers special ? I just found this article on web, which was informative, but was little hard to understand. Can someone explain me this mystery behind prime numbers, any links web resources is much appreciated.
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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."
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Hi, This is basically a math question, but very much applicable to many of the computer algorithms. I know the fact that, - Any integer can be expressed as product of prime factors. - Prime factors can be used to find LCM and HCF - Prime factors can be used to check whether a number divides "N" ( N - Integer ) - Etc.. Etc... My question is.. Why this unique ability for prime numbers ? How is it possible that, any number can be expressed as product of prime factors ? What is it, which makes these prime numbers special ? I just found this article on web, which was informative, but was little hard to understand. Can someone explain me this mystery behind prime numbers, any links web resources is much appreciated.
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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.
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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.
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.
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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.
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.
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Hi, This is basically a math question, but very much applicable to many of the computer algorithms. I know the fact that, - Any integer can be expressed as product of prime factors. - Prime factors can be used to find LCM and HCF - Prime factors can be used to check whether a number divides "N" ( N - Integer ) - Etc.. Etc... My question is.. Why this unique ability for prime numbers ? How is it possible that, any number can be expressed as product of prime factors ? What is it, which makes these prime numbers special ? I just found this article on web, which was informative, but was little hard to understand. Can someone explain me this mystery behind prime numbers, any links web resources is much appreciated.
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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.
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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.
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 primeSo 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.
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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 primeSo 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.
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.
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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.
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.
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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.
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"
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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"
Forget my last posting it is trash. I didn't know what I wrote. Yes, I know what vacuous truth is. And I don't want to raise to question.
harold aptroot wrote:
clearly an empty set contains only prime numbers
That is wrong, because: any set X, the empty set 0 is a subset of X. This is equivalent to asserting that every element of 0 is an element of X, which is vacuously true since there are no elements of 0. (from your link). That doesn't mean it contains only prime numbers. That means that the empty set is part of prime numbers (as it's part of every other set), by meaning of vacuous truth. :)
------------------------------ 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.
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Forget my last posting it is trash. I didn't know what I wrote. Yes, I know what vacuous truth is. And I don't want to raise to question.
harold aptroot wrote:
clearly an empty set contains only prime numbers
That is wrong, because: any set X, the empty set 0 is a subset of X. This is equivalent to asserting that every element of 0 is an element of X, which is vacuously true since there are no elements of 0. (from your link). That doesn't mean it contains only prime numbers. That means that the empty set is part of prime numbers (as it's part of every other set), by meaning of vacuous truth. :)
------------------------------ 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.
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Hey Rage, That was a sharp, and precise reply. Thanks. The link for "Euclid's lemma" was helpful, however it builds other theorems based on the fundamental fact that " Any non prime number can be expressed as product of prime numbers". Lastly, what an explanation..
Rage wrote:
As you have pointed out, their properties can be used in a lot of algorithms. But this is the case for other "type" of numbers having other properties used in other type of algorithms. So your question is no easy to answer... It is like asking why knives are useful to cut something.
I just loved it, But sadly.. This is what I want someone to answer for me.. :thumbsup:
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
Vijay Sringeri wrote:
I just loved it, But sadly.. This is what I want someone to answer for me.. :thumbsup:
Here you go: Knives are useful to cut for their sharp edges :laugh:
To alcohol! The cause of, and solution to, all of life's problems - Homer Simpson ---- Our heads are round so our thoughts can change direction - Francis Picabia
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That is indeed the definition of a prime (a number with no +ive divisors other than 1 and itself). But the OP wrote "Any integer can be expressed as product of prime factors." and 1 is not a prime. /ravi
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