Why prime factorization ?
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harold aptroot wrote:
And it can be, an empty set is a perfectly valid set of prime number, it just happens to be empty.
No! That's wrong by mathematical definitions! By definition the empty set is the unique set having no elements and the axiom of extensionality shows that there is only one empty set. So there is no empty set of prime numbers. There exists only one empty set. No prime numbers at all. :rolleyes:
------------------------------ 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.
The empty set contains 'only prime numbers' in that it doesn't contain any non-primes. If the product of an empty set is defined to be 1, and I think it is, then Harold's statement is true. We're into somewhat abstruse mathematical definition territory here, though.
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The empty set contains 'only prime numbers' in that it doesn't contain any non-primes. If the product of an empty set is defined to be 1, and I think it is, then Harold's statement is true. We're into somewhat abstruse mathematical definition territory here, though.
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.
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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.
Uh, I know what the empty set is. But the statement 'set S contains only thing X' is equivalent to 'set S has no members which are not X': in this case 'the empty set contains only primes' is equivalent to 'the empty set has no non-prime members'. And since it has no members at all, that is clearly true! The empty set also only contains blue items, non-prime items, even numbers, or any other set. Empties are weird like that, a bit like zero being divisible by everything.
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Uh, I know what the empty set is. But the statement 'set S contains only thing X' is equivalent to 'set S has no members which are not X': in this case 'the empty set contains only primes' is equivalent to 'the empty set has no non-prime members'. And since it has no members at all, that is clearly true! The empty set also only contains blue items, non-prime items, even numbers, or any other set. Empties are weird like that, a bit like zero being divisible by everything.
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.
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Ravi Bhavnani wrote:
1 is not a prime
I remember having had to copy this 100 times back when I still was in school. And still don't remember it. Grrrrr.
~RaGE();
I think words like 'destiny' are a way of trying to find order where none exists. - Christian Graus Do not feed the troll ! - Common proverb
It is, I think, fairly common that the trivial object is actually TOO simple. For example, the empty space is not connected, the trivial ring is not a field, and 1 is not a prime number. The reason is an existence-uniqueness one - in this case, the prime factor representation always exists for a number, but it's not unique unless 1 is considered to not be prime.
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Vijay Sringeri wrote:
Why this unique ability for prime numbers ?
Which one exactly ?
Vijay Sringeri wrote:
How is it possible that, any number can be expressed as product of prime factors ?
Vijay Sringeri wrote:
What is it, which makes these prime numbers special ?
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.
~RaGE();
I think words like 'destiny' are a way of trying to find order where none exists. - Christian Graus Do not feed the troll ! - Common proverb
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...[^]
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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...[^]
Thanks all, for showing keen interest in answering/trying to answer this questions. But my question still remains unanswered :doh: However, I just wanted to put some info here. Prime number : Numbers > 1, and which has 1 and itself as it factor is prime nuber. Composite number : All non prime numbers are composite numbers. What about 1 then ? 1 is neither prime nor composite.
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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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...[^]
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
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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...[^]
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[^]
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Thanks all, for showing keen interest in answering/trying to answer this questions. But my question still remains unanswered :doh: However, I just wanted to put some info here. Prime number : Numbers > 1, and which has 1 and itself as it factor is prime nuber. Composite number : All non prime numbers are composite numbers. What about 1 then ? 1 is neither prime nor composite.
Regards, Vijay Blog : Amusement of a speculative mind...[^] Projects : Amusement of a dilettante mind...[^]
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.
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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.
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.
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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.