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Thread: find all solutions to 2p + 1 = n^2, for n natural, p prime

  1. #21
    Elite Member
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    Re: find all solutions to 2p + 1 = n^2, for n natural, p prime

    Quote Originally Posted by twisted_logic89

    ...only odd n values would work right? ....Depends - what do you mean by "work"!
    ... mathematics is only the art of saying the same thing in different words - B. Russell

  2. #22
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    Re: find all solutions to 2p + 1 = n^2, for n natural, p prime

    if n is a natural number and p is a prime number find all solutions to 2p + 1 = n^2

    Since 2p + 1 is always odd, only the odd squares are candidates.

    i.....1.....2.....3.....4.....5...
    n.....1.....3.....5.....7.....9...
    n^2...1.....9....25....49....81...
    p.....0.....4....12....24....40...
    Diff....4.....8.....12....16
    Diff.......4.....4......4
    With the 2nd differences being constant at 4, the general expression for p is of the form p = ai^2 + bi + c

    Using the data above, we can write
    a(1^2) + b(1) + c = 0 or a + b + c = 0
    a(2^2) + b(2) + c = 4 or 4a + 2b + c = 4
    a(3^2) + b(3) + c = 12 or 9a + 3b + c = 12

    Solving, a = 2, b = -2 and c = 0

    Therefore, p = 2i^2 - 2i or p = 2i(i - 1)

    Since p always has a factor of 2, there is no prime number that will satisfy n^2 = 2p + 1.
    TchrWill

    No matter how insignificant it might appear, learn something new every day.

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