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  1. #31
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by Qre:us View Post
    Infinite cannot imply exhaustive as it would negate the whole concept of infinite (boundless). Exhaustive only goes so far as explanation for what a computer by us can generate for a step-by-step sequence, for primes. Thus, it has its human limits. That doesn't mean there's 'limits' to the system of logic on which this numbers game is based. I would disagree with the word exhaustive. I am getting what you mean from a purely syntax model of logic, but, in this scenario, of primes and infinite, it is meaningless. Being or not being exhaustive. That's only relevant in terms of 'if humans wanted to generate all prime numbers'. I think the key question is: 'if humans CAN have the ability to generate all prime numbers'. I.e., the key to the lock.



    Why would we want to find them all? Why would that logically and realistically be a goal of ours in the first place? To find all of them - literally? All prime numbers? I don't understand this premise.

    It is only to find the system by which primes work. To find order in this apparant chaos that is such an anomaly in the field of such rigorous deductive logic that is mathematics.

    Now, if we want to apply mathematical limits to infinite in the treatment of primes, here's a read off Wiki:
    Prime number theorem - Wikipedia, the free encyclopedia
    The reason why you'd want an exhaustive list is that the problem is only paritally solved otherwise. Prime numbers will always be considered something of an elusive mystery until one can show how they all can be found. Once they are all found then the mystery is totally solved.
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  2. #32
    Senior Member Qre:us's Avatar
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    Quote Originally Posted by noigmn View Post
    Also if you can find a pattern in prime numbers, great! But I 'feel' there shouldn't be one.
    Feeling aside, this may be the resolution to this 'problem'. However, only if we can answer: What would logically make us conclude that prime numbers occur at random?

    We are using numbers to solve numbers. Maybe we need to use something else for once.
    Like? Full understanding of the properties of a random process? And, that, by the play of statistics, within randomness, we can still find parts that may look like a pattern/order? Which doesn't necessarily *make* it so?

  3. #33
    Senior Member Qre:us's Avatar
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    Quote Originally Posted by The_Liquid_Laser View Post
    Prime numbers will always be considered something of an elusive mystery until one can show how they all can be found.
    Showing how they can all be found, and actually doing it, are different matters. One gets to the base, the other exercises the propositions on which the base is built. That is why 'showing how they can all be found' is of interest to me, and why I cannot understand the relevance of actually, doing it...finding them all. Then, going backwards and saying, oh, that was the process by which it was achieved. It's a losing battle even before you start. As you will never find them all, as they are all infinite.

    Once they are all found then the mystery is totally solved.
    That's like saying that Pi is still unsolved because we haven't found the 'last number' of its decimal representation - well, that's cuz it never ends. But, this is not the case. We can predict how to get more and more accurate decimal representations of Pi, but, we'll never reach its 'end'. (as it is a trancendental number).

  4. #34
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by Qre:us View Post
    Showing how they can all be found, and actually doing it, are different matters. One gets to the base, the other exercises the propositions on which the base is built. That is why 'showing how they can all be found' is of interest to me, and why I cannot understand the relevance of actually, doing it...finding them all. Then, going backwards and saying, oh, that was the process by which it was achieved. It's a losing battle even before you start. As you will never find them all, as they are all infinite.



    That's like saying that Pi is still unsolved because we haven't found the 'last number' of its decimal representation - well, that's cuz it never ends. But, this is not the case. We can predict how to get more and more accurate decimal representations of Pi, but, we'll never reach its 'end'. (as it is a trancendental number).
    Heh, I think you just misunderstand what I mean. For example we know what all of the natural numbers are even though we can't physically write them all down. That is what I mean for the primes.
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  5. #35

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    Quote Originally Posted by Qre:us View Post
    Feeling aside, this may be the resolution to this 'problem'. However, only if we can answer: What would logically make us conclude that prime numbers occur at random?



    Like? Full understanding of the properties of a random process? And, that, by the play of statistics, within randomness, we can still find parts that may look like a pattern/order? Which doesn't necessarily *make* it so?
    They don't occur at random. That is why I put feel, I can't logically back it up. They have a very obvious order. They are the spaces in this.

    (2),4,6,8,10,12,14,16
    (3),6,9,12,15
    (5),10,15
    (7),14
    (11)
    (13)

    The thing is you need the multiplication tables to see the spaces. It is not random, it is just of a complexity that seems impossible to simplify. You get the improvement of removing redundant information, ie. 6,10,12,14,15 are mentioned twice, so you only need to check for 2 and 3 as factors, but you can't remove the underlying complexity, which is that you are finding spaces in a mesh made from multiplying prime numbers, and without accounting for all these possible options, you can't know it is prime. I am unaware of any method that doesn't just identify more redundant information. Eventually you hit entropy, if we haven't already. You won't simplify the problem of checking any further.

    The patterns people find are there, but are normally the equivalent of checking for certain primes. They are just another check in disguise, and take the same number of calculation steps as checking for those primes would. There are many maths problems that people see as holy grails in solving it. But they are still the same problem dressed up enough that people are willing to play with numbers for years rather than break it down to the question of what are we doing and saying. You've got as little chance of solving them as the original problem. And unless philosophically the problem is suddenly seen in a way that avoids redoing it over and over in the same style of thinking, I don't believe it is solvable.
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  6. #36

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    I've been sleeping a lot.

    But I should mention some functions of extreme importance to primes mentioned in my number theory class.

    Totient Function -- from Wolfram MathWorld

    The totient function of a prime number, p, is p-1

    A more complex relation involves the Riemann zeta function and a particular Euler product.

    Studying primes is a very deep subject.

    Accept the past. Live for the present. Look forward to the future.
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  7. #37

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    Quote Originally Posted by ygolo View Post
    I've been sleeping a lot.

    But I should mention some functions of extreme importance to primes mentioned in my number theory class.

    Totient Function -- from Wolfram MathWorld

    The totient function of a prime number, p, is p-1

    A more complex relation involves the Riemann zeta function and a particular Euler product.

    Studying primes is a very deep subject.
    cool, I've seen them both before, but not read that deeply about them, how do they help? Like once you strip all the transforms and maths away, what is their value? What can they tell us about the problem?

    Like it is all something I'm interested in, but a bit cynical about too. I'd prefer to just be the first part .
    Freude, schöner Götterfunken Tochter aus Elysium, Wir betreten feuertrunken, Himmlische, dein Heiligtum! Deine Zauber binden wieder Was die Mode streng geteilt; Alle Menschen werden Brüder, Wo dein sanfter Flügel weilt.

  8. #38

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    Quote Originally Posted by noigmn View Post
    cool, I've seen them both before, but not read that deeply about them, how do they help? Like once you strip all the transforms and maths away, what is their value? What can they tell us about the problem?

    Like it is all something I'm interested in, but a bit cynical about too. I'd prefer to just be the first part .
    Well, calculating the totient function is seen as an alternative to factorization. That's kind of cool I think. In a way, you are solving the same problem as factorization.

    The Reimann Zeta function connects so many branches of mathematics, its har to know where to begin. It's just cool. Look at the simplicity of it.

    Euler proved that Zeta function equaled a particular product of functions of primes. Which really allows us to get an idea of the density of prime numbers.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  9. #39

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    Quote Originally Posted by ygolo View Post
    Well, calculating the totient function is seen as an alternative to factorization. That's kind of cool I think. In a way, you are solving the same problem as factorization.

    The Reimann Zeta function connects so many branches of mathematics, its har to know where to begin. It's just cool. Look at the simplicity of it.

    Euler proved that Zeta function equaled a particular product of functions of primes. Which really allows us to get an idea of the density of prime numbers.
    Yeh, Euler's part is almost intuitive. I think one time, I rederived it accidentally. It states something that is intrinsic to what a prime is. Which is what worries me about the zeta function. Could the Euler stuff more disprove the value of the zeta function's use with primes than support it. Like once you see the zeta function that way, does it really say much that can help with finding primes. And if it does, does this support the idea that it is not solvable, rather than the idea it can solve something.
    Freude, schöner Götterfunken Tochter aus Elysium, Wir betreten feuertrunken, Himmlische, dein Heiligtum! Deine Zauber binden wieder Was die Mode streng geteilt; Alle Menschen werden Brüder, Wo dein sanfter Flügel weilt.

  10. #40

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    Quote Originally Posted by noigmn View Post
    Yeh, Euler's part is almost intuitive. I think one time, I rederived it accidentally. It states something that is intrinsic to what a prime is. Which is what worries me about the zeta function. Could the Euler stuff more disprove the value of the zeta function's use with primes than support it. Like once you see the zeta function that way, does it really say much that can help with finding primes. And if it does, does this support the idea that it is not solvable, rather than the idea it can solve something.
    Oh. I think the zeta function is incredibly important. You can look at the following for insight. It is really important to the distribution of primes.

    On the Number of Primes Less Than a Given Magnitude - Wikipedia, the free encyclopedia

    Here is a pdf of the translation:
    http://www.maths.tcd.ie/pub/HistMath...Zeta/EZeta.pdf

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

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