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  1. #1
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    urgent

    can you please help?

    Find all twin primes p, p+2 whose mid term p+1 is i) Triangular ii) Perfect square iii) perfect


    thx
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  2. #2
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    Quote Originally Posted by marlen19861@hotmail.com View Post
    can you please help?

    Find all twin primes p, p+2 whose mid term p+1 is i) Triangular ii) Perfect square iii) perfect


    thx
    i) If p+1 is a triangular number there exists a natural number k such that:

    p+1=\frac{k(k+1)}{2}

    so:

    k^2+k-(2p+2)=0

    and hence:

    k=\frac{-1 \pm \sqrt{p+2}}{2}

    But the left hand side is an integer and the right hand side is irrational if p+2 is not a perfect square, but p+2 is prime. Hence there exist no twin priimes whose mid term is a triangular number.

    CB
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  3. #3
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    is the last question correct?
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  4. #4
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    Quote Originally Posted by nikolany View Post
    is the last question correct?
    Now that I think about it it can't be since 5,6,7 would be a counter example.

    The last equation should be:

    k=\frac{-1\pm\sqrt{8p+9}}{2}

    Then we require that 8p+9 be a perfect square for both sides to be rational, which does not obviously work

    CB
    Last edited by CaptainBlack; November 10th 2008 at 12:20 AM.
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  5. #5
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    Quote Originally Posted by CaptainBlack View Post
    Now that I think about it it can't be since 5,6,7 would be a counter example.

    The last equation should be:

    k=\frac{-1\pm\sqrt{8p+9}}{2}

    Then we require that 8p+9 be a perfect square for both sides to be rational, which does not obviously work

    CB
    Having thought about this some more I can now make this work, but I won't go into the detail of the solution as Opalg's solution in the other thread is much neater.

    But the gist of the solution is that we can reduce the condition that 8p+9 is a perfect square to: there exists a positive integer k such that

    2p=(k-1)(k+2)

    which by the fundamental theorem of arithmetic forces us to conclude p=??.

    CB
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