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Web--of-- Ones polynomials derived(1)

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  • aldrich617
    Begin with a quadratic equation whose solutions have prime factors limited to integers ending in 1,5 or 9 such as x^2 + x -1, generating the sequence
    Message 1 of 1 , Mar 18, 2006
      Begin with a quadratic equation whose solutions have prime factors
      limited to integers ending in 1,5 or 9 such as x^2 + x -1,
      generating the sequence 1,5,11,19,29,41,55,71,89,109,131,155...
      Find a sub sequence whose members all end in one such as
      5 * (11,31,61,101,151) : five are sufficient to generate the rest
      using difference equations. With the square version of the
      ancient form of Ulam's spiral produce a partial grid such as :

      3251 3001 2761 2531 2311
      1361 1201 1051 911 2101
      451 361 281 781 1901
      101 61 211 661 1711
      11 31 151 551 1531

      Check all of the diagonals left to right, bottom to top,
      for prime factors that
      end with a nine within twenty or so terms and eliminate them from
      consideration; We are left with two sequences, both of which
      are good candidates for inclusion in the Web-of Ones:
      11, 61, 281, 911, 2311... and
      31, 211,781 2101, 4651....
      From the central diagonal check the rows and columns similarly,
      11, 101, 451, 1361 3251 looks good. There may be more usable
      sequences in the extended grid. A rule analagous to a primality
      test to checkthese sequences without actually factoring all
      of its members is needed, because such a task is impossible.
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