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Math Help - Goldbach's Conjecture

  1. #1
    Grand Panjandrum
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    Goldbach's Conjecture

    We know that the strong Goldbach conjecture:

    "Every even number \ge 4 is the sum of two primes"

    is an open question with a price on its head.

    What I want to know is has a resticted version of this been proven for
    special types of positive even integers, say the squares, cubes, whatever?

    RonL
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  2. #2
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    Quote Originally Posted by CaptainBlank View Post
    What I want to know is has a resticted version of this been proven for
    special types of positive even integers, say the squares, cubes, whatever?
    Unsolved problem sent as a letter to Euler in 1742.

    The numerical results are astonding, for example there are 219,400 such representations for 100,000,000. And in fact it seems as the number of possible representations keeps increasing. And there is nobody who seriously believes that this conjecture is false.

    It is know that each even integer is a sum of six or fewer primes.

    The first real progess was made by Hardy and Littlewood in 1922, they have shown that for a sufficiently large integer the Conjecture of Goldbach is true! But the sad news is that Hardy and Littlewood used the Generalized Riemann Hypothesis. In 1937 the Russian mathematician Vinogradov was to remove the necessatity of using the generalized Riemann Hypothesis and established that all sufficiently large integers are some of three odd primes. But he was unable to decide how large this number is. In 1956 Borozdkin proved that 3^{3^{15}} is big enough. And in 1989 this number was reduced to 10^{4300}. Hence if it can be checked by hand that all of those numbers are true, then the Goldbach Conjecuture is settled. But it is far to difficult for a man to do. And even if we accept computer results (because I do not accept anything that a computer does and many mathematicians agree) it is still computationally difficult.


    Note: I have taken the above facts from my Number Theory book. I have changed the wording to avoid copyright material.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    Unsolved problem sent as a letter to Euler in 1742.

    The numerical results are astonding, for example there are 219,400 such representations for 100,000,000. And in fact it seems as the number of possible representations keeps increasing. And there is nobody who seriously believes that this conjecture is false.

    It is know that each even integer is a sum of six or fewer primes.

    The first real progess was made by Hardy and Littlewood in 1922, they have shown that for a sufficiently large integer the Conjecture of Goldbach is true! But the sad news is that Hardy and Littlewood used the Generalized Riemann Hypothesis. In 1937 the Russian mathematician Vinogradov was to remove the necessatity of using the generalized Riemann Hypothesis and established that all sufficiently large integers are some of three odd primes. But he was unable to decide how large this number is. In 1956 Borozdkin proved that 3^{3^{15}} is big enough. And in 1989 this number was reduced to 10^{4300}. Hence if it can be checked by hand that all of those numbers are true, then the Goldbach Conjecuture is settled. But it is far to difficult for a man to do. And even if we accept computer results (because I do not accept anything that a computer does and many mathematicians agree) it is still computationally difficult.


    Note: I have taken the above facts from my Number Theory book. I have changed the wording to avoid copyright material.
    Yea, yea, I know all that, if I could find the information I'm after in an
    easily accessible souce I would not be asking

    (By the way the text of Goldbach's letter is available here)

    RonL
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    Quote Originally Posted by CaptainBlank View Post

    (By the way the text of Goldbach's letter is available here)
    Look interesting.

    So Euler knew German?

    Can somebody please translate (by somebody I mean Earboth).
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    Look interesting.

    So Euler knew German?

    Can somebody please translate (by somebody I mean Earboth).
    what about this:




    RonL
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  6. #6
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    Hard to read, but it seems that Euler is using an infinite series in his approach somewhere.
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  7. #7
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    Hi everyone,
    any one to help me to know if there are still some reward for someone who prove the goldbach conjecture,
    Many thanks.
    (PS. I know it was some reward until march 2002...)
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  8. #8
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    Attempted proof of Goldbach's Conjecture

    I have an attempted proof of Goldbach's Conjecture. Does anyone have any advice on how to get it reviewed?
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    Quote Originally Posted by ProfessorB View Post
    I have an attempted proof of Goldbach's Conjecture. Does anyone have any advice on how to get it reviewed?
    If you indeed have enough knowledge of mathematics to solve this problem then you would not not be asking where you can get it reviewed.
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  10. #10
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    Thank you for your kind help. Your obvious superior knowledge has produced in me a hitherto unequaled humility. It goes without saying, of course, that you could not possibly be wrong. That condition is reserved for the merely human.
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  11. #11
    Grand Panjandrum
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    Quote Originally Posted by ProfessorB View Post
    I have an attempted proof of Goldbach's Conjecture. Does anyone have any advice on how to get it reviewed?
    Write it up as a paper and submit it to a refereed mathematical journal. Of course they will not even look at it if it does not conform to accepted convention for maths papers. Even then they may well not look at it if you do not have an identifiable mathematical background (journals receive so may erroneous proofs of GC and FLT that they often do not even look at such papers unless the author has credentials and or background).

    Alternatively you could post it on ArXiv.org where you may get some feedback.


    CB
    Last edited by CaptainBlack; October 12th 2008 at 08:25 PM.
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    Thank you for responding. ArXive.org is, however, not available (under construction). I have posted it to a non-mathematical forum but don't have a lot of optimism for response.

    Again, thanks for your advice.
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  13. #13
    Grand Panjandrum
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    Quote Originally Posted by ProfessorB View Post
    Thank you for your kind help. Your obvious superior knowledge has produced in me a hitherto unequaled humility. It goes without saying, of course, that you could not possibly be wrong. That condition is reserved for the merely human.
    You should take note of what ImPerfectHacker says for a number of reasons, one of which is that mathematical background will be taken into account by anyone qualified to read your proof before they bother.

    CB
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  14. #14
    Grand Panjandrum
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    Quote Originally Posted by ProfessorB View Post
    Thank you for responding. ArXive.org is, however, not available (under construction). I have posted it to a non-mathematical forum but don't have a lot of optimism for response.

    Again, thanks for your advice.
    ArXiv.org seems to be working for me (note spelling mistake in site name in earlier post it seems the other spelling has been hijacked).

    CB
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  15. #15
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    Again, I appreciate your advice. I realize the improbability of my claim - ImPerfectHacker just has a rather boorish way of communicating.

    I may indeed be wrong, but I am not without some justification in believing I have, if not having solved GC, then at least having made some progress. I am also aware that history has provided us with some examples of progress being made in mathematics by those without formal education in the field - Ramanujan being the most striking example.

    Once again, thank you.
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