Loading ...
Sorry, an error occurred while loading the content.

Prime Chain of 128

Expand Messages
  • aldrich617
    This is a Prime Chain of 128 terms, including 104 distinct primes, consisting of the output of eight equations that alternate sequentially within a procedural
    Message 1 of 1 , Jul 12, 2008
    • 0 Attachment
      This is a Prime Chain of 128 terms, including
      104 distinct primes, consisting of the output of
      eight equations that alternate sequentially within
      a procedural expression of a single polynomial.
      The equations are either subsequences of
      x^2 - 79x + 1601 or transforms
      with one exception: 100x^2 -2260x + 12959

      The other four distinct equations are
      Euhler-derived.:
      25x^2 -1185x + 14083,
      25x^2 -775x + 6047,
      100x^2 -2280x + 13159,
      100x^2 -4160x +43427

      14083, 14087, 6047, 13159, 12923, 43427,
      5297, 12959, 11813, 11827, 4597,
      10979, 10753, 39367, 3947, 10799,
      9743, 9767, 3347, 8999 ,8783,
      35507, 2797, 8839, 7873, 7907,
      2297, 7219, 7013, 31847, 1847,
      7079, 6203, 6247, 1447, 5639
      5443, 28387, 1097, 5519, 4733
      4787, 797, 4259 4073, 25127,
      547, 4159, 3463, 3527, 347,
      3079, 2903, 22067, 197, 2999,
      2393, 2467, 97, 2099, 1933,
      19207, 47, 2039, 1523, 1607,
      47, 1319, 1163, 16547, 97,
      1279, 853, 947, 197, 739,
      593, 14087, 347, 719, 383,
      487, 547, 359, 223, 11827,
      797, 359, 113, 227, 1097,
      179, 53, 9767, 1447, 199,
      43, 167, 1847, 199, 83,
      7907, 2297, 239, 173, 307,
      2797, 419, 313, 6247, 3347,
      479, 503, 647, 3947, 839,
      743, 4787, 4597, 919, 1033,
      1187, 5297, 1459, 1373, 3527,
      6047, 1559

      Perhaps its obvious, but I think that I should mention that the
      basis for this prime chain in my little procedure at bottom is
      the difference equation stack concept. This allows one to
      generate all of the values of a polynomial without knowing the
      equation providing the first few outputs are known. A simple example
      can be illustrated. For instance, if one has the outputs 41, 43, 47
      then a difference equation pyramid can be developed that will
      generate the same sequence as x^2 - x + 41:
      2
      2 4
      41 43 47
      Using my procedure, I would begin with the stack on the
      left-hand side of the pyramid and start a "repeat loop".
      2 2 2 2
      2 + 2 = 4 + 2 = 6 + 2 = 8
      41 + 2 = 43 + 4 = 47 + 6 = 53 etc.
      A more complex difference equation pyramid is necessary to
      generate the rest of the sequence for 11, 61, 281, 911,2311...
      (5x^4 - 10x^3 + 20x^2 -15x + 11). The procedure is easily
      adjustable to do this for any degree polynomial.
      120
      240 360
      170 410 770
      50 220 630 1400
      11 61 281 911 2311
      Sarting with the left-hand stack as before, the calculations proceed:
      120 120 120 120
      240 + 120 = 360 + 120 = 480 + 120 = 600
      170 + 240 = 410 + 360 = 770 + 480 = 1250
      50 + 170 = 220 + 410 = 630 + 770 = 1400
      11 + 50 = 61 + 220 = 281 + 630 = 911 etc.

      I believe that concept of the difference equation stack probably
      predates polynomial nomenclature, though my number theory books
      fail to mention the idea at all. The melding of several polynomials
      at once with a difference equation stack is not mentioned either,
      but it may be easily observed that success in that endeavor will
      surely depend heavily on the initial pieces of equations and
      transforms chosen, and their ordering, and that certainly,without
      modification, the difference equation pyramid will always go to
      infinity in that situation and be useless.

      I have found that the use of the sign change of the absolute value
      function on selected rows can often be salutory in that regard,
      and resolve the pyramid to a constant at its apex.
      At each row from bottom to top one must experiment with changing
      all negative terms to positive with an eye to causing the next row
      to be smaller in size, less erratic in pattern, and for the last
      digit of all terms to be identical. There may be several ways to get
      the pyramid to resolve to a constant,or none at all, and it is
      quite possible that there are smaller initial left-hand stacks for
      our present polynomial (abs denotes rows to be modified by the
      absolute value function)
      than this one:
      abs 160
      0
      abs 1820
      0
      abs 1760
      abs 0
      abs 59160
      abs 0
      -1090
      abs 125806
      62518
      abs 29414
      16284
      -696
      abs 23196
      -804
      4
      14083

      The Pascal procedure below should run as an import in several
      programming
      environments as is.

      procedure Ndegrees5;
      var a : array[0..32] of extended;
      ct: longint;
      n,nh ,i,j : integer;
      ab1,ab2 : extended;
      begin
      for i := 0 to 32 do
      a[i] := 0;
      N := 17;
      a[0] := 14083{ FIRST TERM OF PRIME CHAIN};
      writeln('1');
      writeln(trunc(a[0]));
      writeln;
      nh := 1;
      a[1] := 14087 ;a[2] := 6047;
      a[3] := 13159 ; a[4] :=12923 ; a[5] := 43427; a[6] := 5297 ;
      a[7] := 12959 ; a[8] :=11813 ; a[9] :=11827 ;a[10] := 4597 ;
      a[11] :=10979 ; a[12] := 10753 ; a[13] := 39367 ;a[14] := 3947;
      a[15] :=10799 ;a[16] := 9743 ;a[17] := 9767 ;
      repeat
      for i := N downto nh do
      begin
      a[i] := a[i] - a[i-1] ;
      IF NH = 3 THEN A[I] := abs(A[I]); {}
      iF NH = 6 THEN A[I] := abs(A[I]); {}
      iF NH = 8 THEN A[I] := abs(A[I]); {}
      iF NH = 10 THEN A[I] := abs(A[I]); {}
      iF NH = 11 THEN A[I] := abs(A[I]); {}
      iF NH = 12 THEN A[I] := abs(A[I]); {}
      iF NH = 13 THEN A[I] := abs(A[I]); {}
      iF NH = 15 THEN A[I] := abs(A[I]); {}
      iF NH = 17 THEN A[I] := abs(A[I]); {}
      end;
      nh := nh + 1;
      until nh = n + 2;
      ct := 0;
      repeat
      ct := ct + 1;
      ab1 := a[n] + a[n-1];
      for i := N-1 downto 1 do
      begin
      if i = 15 then if ct mod 2 = 0 then a[15] := -a[15];{}
      if i = 14 then if ct mod 4 = 0 then a[14] := -a[14];{}
      if i = 13 then if ct mod 2 = 0 then a[13] := -a[13];{}
      if i = 12 then if ct mod 4 = 0 then a[12] := -a[12];{}
      if i = 11 then if ct mod 2 = 0 then a[11] := -a[11];{}
      if i = 10 then if ct mod 8 = 2 then a[10] := -a[10];{}
      if i = 10 then if ct mod 8 = 0 then a[10] := -a[10];{}
      if i = 8 then if ct mod 2 = 1 then a[8] := -a[8];{}
      if i = 6 then if ct mod 4 = 2 then a[6] := -a[6];{}
      if i = 6 then if ct mod 4 = 3 then a[6] := -a[6];{}
      if i = 3 then if ct mod 2 = 0 then a[3] := -a[3];{}

      ab2 := a[i] + a[i-1] ;
      a[i] := ab1;
      ab1 := ab2;
      end;
      if odd(ct) then a[17] := -a[17];

      a[0] := ab1;
      writeln(ct + 1);
      writeln(trunc(a[0]));{}
      READLN;
      until 1<0;

      Aldrich Stevens
    Your message has been successfully submitted and would be delivered to recipients shortly.