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Re: [PrimeNumbers] Re: A Property

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  • Jacques Tramu
    - I have designed a new (i think it is new) and quite simple compression algorithm which can compress a string of ANY length into a string of 666 bytes (using
    Message 1 of 64 , Oct 21, 2005
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      - I have designed a new (i think it is new) and quite simple compression
      algorithm which can compress a string of ANY length into a string
      of 666 bytes (using the well-known fact that 36 n^2 - 666 n + 1277 is prime
      for all n between 0 and 42,inclusive )(1)

      - I'm currently working on the corresponding decompression algorithm ...



      (1)
      -> for(n=0;n<=42;n++) {p = 36*n**2 - 666*n +1277; println(n,p,isprime(p));}

      0 1277 2
      1 647 2
      2 89 2
      3 -397 2
      4 -811 2
      5 -1153 2
      6 -1423 2
      7 -1621 2
      8 -1747 2
      9 -1801 2
      10 -1783 2
      11 -1693 2
      12 -1531 2
      13 -1297 2
      14 -991 2
      15 -613 2
      16 -163 2
      17 359 2
      18 953 2
      19 1619 2
      20 2357 2
      21 3167 2
      22 4049 2
      23 5003 2
      24 6029 2
      25 7127 2
      26 8297 2
      27 9539 2
      28 10853 2
      29 12239 2
      30 13697 2
      31 15227 2
      32 16829 2
      33 18503 2
      34 20249 2
      35 22067 2
      36 23957 2
      37 25919 2
      38 27953 2
      39 30059 2
      40 32237 2
      41 34487 2
      42 36809 2
      -------------------------------------
      http://www.echolalie.com
      -------------------------------------
    • Kermit Rose
      ... Thank you David. I had been confused by consideration of the following: if x y = z and y x, then if we set t = (y+x)/2 and s = (y-x)/2 then z = t^2 - s^2
      Message 64 of 64 , Oct 20, 2012
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        On 10/20/2012 10:07 AM, primenumbers@yahoogroups.com wrote:
        > 2.2. Re: Question
        > Posted by: "djbroadhurst"d.broadhurst@... djbroadhurst
        > Date: Fri Oct 19, 2012 9:02 am ((PDT))
        >
        >
        >
        > --- Inprimenumbers@yahoogroups.com,
        > Kermit Rose<kermit@...> wrote:
        >
        >> >Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
        >> >
        >> >for what values of x would
        >> >x^2 + 5 x + 6 be a perfect square.
        > Set A = 1, B = 0, C = -1, D = 5, E = 0, F = 6.
        >
        > and you will get the obvious answer from Dario:
        >
        > x = -2
        > y = 0
        > and also:
        > x = -3
        > y = 0
        > Calculation time: 0h 0m 0s
        >
        > David

        Thank you David.

        I had been confused by consideration of the following:

        if x y = z and y > x,
        then if we set t = (y+x)/2 and s = (y-x)/2

        then z = t^2 - s^2
        and x = y - 2 t.

        So z = x y = (y - 2t) y = y^2 - 2 t y

        y^2 - 2 t y - z = 0

        Transforming w = y - k yields

        (w + k)^2 - 2 t (w + k) - z = 0

        w^2 + (2 k - 2 t) w + (k^2 - 2 t k - z) = 0

        which has integral solution if and only if

        (2k - 2 t)^2 + (z + 2 t k - k^2) is an integral square

        In all that derivation, I forgot that I still did not know what value t was.

        I confused myself too easily.

        Kermit








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