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Math Help - prime number proof help

  1. #1
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    prime number proof help

    prove tha there is no prime number p such that  p^{2} -81 is divisible by 54.


    suppose  54/ p^{2} - 81

    than

     p^{2} -81 = 54K for some K in the naturals.


    how would I proceed from here? very confused, I am suppose to come up with some sort of contradiction.

    any help appreciated.
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  2. #2
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    Re: prime number proof help

    Look at it the "other way around": p^2= 54K- 81= 9(6- k)
    Thanks from Tweety and topsquark
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  3. #3
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    Re: prime number proof help

    Quote Originally Posted by HallsofIvy View Post
    Look at it the "other way around": p^2= 54K- 81= 9(6- k)

    Not sure what you mean....

    so  p^{2} = 9(6-k)

    hence 9 divides p^2 ?

    so 9 divides p ?
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  4. #4
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    Re: prime number proof help

    if p=9(mod54), p2=81(mod54)

    p-9=kx54
    p=9(1+6k)
    p can't be prime.
    Last edited by Hartlw; November 19th 2013 at 09:46 AM. Reason: mistake
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  5. #5
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    Re: prime number proof help

    Quote Originally Posted by Tweety View Post
    prove tha there is no prime number p such that  p^{2} -81 is divisible by 54.


    suppose  54/ p^{2} - 81

    than

     p^{2} -81 = 54K for some K in the naturals.


    how would I proceed from here? very confused, I am suppose to come up with some sort of contradiction.

    any help appreciated.
    p2=81+54k
    p2=9(9+6k)
    p=3(9+6k)1/2

    so p can't be prime even if (9+6k)1/2 is an integer.
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  6. #6
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    Re: prime number proof help

    Quote Originally Posted by Tweety View Post
    Not sure what you mean....

    so  p^{2} = 9(6-k)

    hence 9 divides p^2 ?

    so 9 divides p ?
    No, either 9 divides p (if 9 also divides 6-k) or 3 divides p. Either way, p is not prime.
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