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Primality and Dr. B's Formula cont'd

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  • aldrich617
    As per request, at the bottom of this message is a sample of equations that can be derived from Dr. B s Fabulous Fibonacci Formula. I have never seen this
    Message 1 of 1 , Jan 15, 2009
      As per request, at the bottom of this message is a sample
      of equations that can be derived from Dr. B's Fabulous
      Fibonacci Formula. I have never seen this formula or
      anything like it appear in a number theory text, yet I believe
      that the topic is significant, and should get more attention.
      It is highly structured infinite family of forms in which
      every prime of every term ends in one, and it is bound to be
      more than a novelty.

      The primality conjectures that I displayed on the Prime Numbers
      website on 12/30/08 concerning this formula are not difficult in
      essence, but they are quite hard to explain clearly, and so I
      have made a visual picture of a small part of 5x^2 +65x + 121
      (the embedding quadratic of 5X^4 - 10x^3 +70x^2 - 65x + 121
      from Dr. B's list) in figure 1 below along with the values
      of other relevant number patterns,hoping to communicate the
      meaning of the numerous simple calculations needed to establish
      the necessary relationships.

      x, A(x), f, T(f), g : integers;
      A(x) = 5x^2 + 65X + 121 ;
      T(f) = 5f^2 - 4A(x) ;

      figure 1 (note: some non-square values of T(f) omitted)

      x A(x) f T( f + 2g) with 0 <= g < 7 + x - x div 2
      -------------------------------------------------------
      0 121 15, 641 31^2 1321 - - - 3161 61^2
      1 191 17, 681 1041 1441 - - - 3441 4041 4681
      2 271 19, 721 1121 1561 - - - 61^2 4361 71^2
      3 361 21, 761 1201 41^2 - - - 4001 4681 5401 6161
      4 461 23, 801 1281 1801 - - - 4281 5001 5761 81^2
      5 571 25, 29^2 1361 1921 - - - 4561 5321 6121 6961 7841
      6 691 27, 881 1441 2041 - - - 841 5641 6481 7361 91^2
      7 821 29, 921 39^2 2161 - - - 5121 5961 6841 7761 8721 9721
      8 961 31, 31^2 1601 2281 - - - 5401 6281 7201 8161 9161 101^2
      9 1111 33,1001 41^2 49^2 - - - 5681 6601 7561 8561 9601 10681 11801

      It is evident that the last number T(f) on every even row is
      a square, a pattern that visually forms a diagonal that is
      continuous (and infinite). A second such diagonal begins at
      the first element of T(f) at x = 5 on all of the odd rows. The
      conjecture of 12/30/08 says that beginning on the row
      that is one above the start of the lower diagonal (here at x =4),
      the value of A(x) on rows that have exactly one square within
      the required number of terms are always prime, and otherwise
      are always composite. Dr. Broadhurst partially proved this (and
      extended the idea quite a bit)for the similar form 5x^2 + 15x + 11,
      a quadratic that can also be derived from his formula, and I
      believe that a general proof may be possible for absolutely all
      analogous equations derived from it. If anyone can develop such
      a proof or find a valid counterexample, kindly share it. Likewise
      for a closely related conjecture also regarding 5x^2 + 15x + 11
      that was proved by Dr. Oakes with a unique factorization in a
      posting of 10/06/08.

      Here I wish to to take the similar and typical 5X^2 +65x +121 from
      above and articulate a way to set it up so that perhaps another
      unique factorization may be performed. The situation is a bit more
      complex, and so such an elegant solution may not be possible,
      but I hope that at least the method of stating the problem will
      be applicable to any equation generated by Dr. B's formula with
      only minor variation.

      Begin as before by locating the lower and upper diagonals of
      squares. At x = 4, count the number of terms of f(k) to the upper
      diagonal (10 terms) and find the square root of T(f) on that
      diagonal (integer R1= 81). Find the square root of the first T(f )
      on row 0 (sqrt 641 = 25+). The smallest permissible integer value
      R2 of the upper diagonal is the first value +/- 1 mod 10 > 25 ,
      {R2 = 31) ,and the new starting point for the equation is
      x = 4 - 2*(R1 - R2)/10 = -6 whose length L1 = 10 - (R1 - R2)/10 = 5.
      Figure 1 is recalibrated into figure 2 below. The other values
      necessary for the new primality conjectures are then easy to see
      or calculate.


      x, A(x), f, T(f), g, x1, x2, f1, f2,
      u1, u2, v1, v2, w, L1, L2,
      b1, b2, S1, S2, d1, d2 : integers;
      A(x) = 5x^2 + 5X -89 ;
      T(f) = 5f^2 - 4A(x) ;

      x1 = 0, A(0) = -89, f1 = 3, T(3) = 401, T(5) = 481,
      u1 = (401 - 1)/40 = 10, v1 = (481 -401)/40 = 2,
      w = u1 - v1 + 1 = 9, L1 = 5, b1 = 31,
      S1 = (b1^2 -1)/40 = 24;

      x2 = 11, A(11) = 571, f2 = 25, T(25) = 29^2, T(27)=1361,
      u2 = (841 - 1)/40 = 21, v2 = (361 -841)/40 = 13,
      L2 = 1, b2 = 29,
      S2 = (b2^2 -1)/40 = 21;

      Also d1 = x2 - x1 = 11 and d2 = L1 - L2 = 4.

      For any even value x1 > 9, let n = L1 + x1/2,
      k1 = x1 + v1 -1 = x1 + 1,and k1..kn be consecutive
      integers with S1 - w = S1 - 9 = k1 + k2..+kn.
      For each S1 find any shorter runs of the form
      S1 -9 = m1 +m2..+ mp, with p < n, m1 > k1.

      Examples:
      x1 = 10, k1 = 11, n = 10,
      S1 - 9 = 155 = 11+12+13+14+15+16+17+18+19+20
      = 29+30+31+32+33
      = 77+78
      = 155
      x1 = 14, k1 = 15, n = 12,
      S1 - 9 = 246 = 15+16+17+18+19+20+21+22+23+24+25+26
      = 60+61+62+63
      = 81+82+83
      = 246

      For any odd value x2 > 10, let n = L1 + (x2 - d1)/2 =
      L1 + (x2 - 11)/2 , i = n - d2 = n-4, k1 = x2 + 1, and
      k1..ki be consecutive integers with S2 - 9 = k1 + k2..+ki.
      For each S2 find any shorter runs of the form
      S2 - w = S2 -9 = m1 +m2..+ mp, with p < n, m1 > k1.
      For each S2 find also any longer runs of the form
      S2 -9 = m1 +m2..+ mp, with i < p <= n, m1 < k1.

      Examples:
      x2 = 21, k1 = 22, n = 10, i = 6,
      S2 - 9 = 147 = 22+23+24+25+26+27
      = 48+49+50
      = 73 + 74
      = 147
      = 18+19+20+21+22+23+24

      x2 = 23, k1 = 24, n = 11, i = 7,
      S2 - 9 = 189 = 24+25+26+28+28+29 +30
      = 29+30+31+32+33+34
      = 62+63+64
      = 94+95
      = 189
      = 17+18+19+20+21+22+23+24+25

      figure 2 (note: some non-square values of T(f) omitted)

      x A(x) f T(f + 2g) with 0 <= g <= 4 + x - x div 2
      -------------------------------------------------------
      0 -89 3, 401 481 601 - 31^2
      1 -79 5, 21^2 561 721 - - 1441
      2 -59 7, 481 641 29^2 - - 41^2
      3 -29 9, 521 721 31^2 - - 1921 2321
      4 11 11, 561 801 1081 - - 2161 51^2
      5 61 13, 601 881 1201 - - 49^2 2881 3401
      6 121 15, 641 31^2 1321 - - 2641 3161 61^2
      7 191 17, 681 1041 1441 - - 2881 3441 4041 4681
      8 271 19, 721 1121 1561 - - 3121 61^2 4361 71^2
      9 361 21, 761 1201 41^2 - - 3361 4001 4681 5401 6161
      10 461 23, 801 1281 1801 - - 601 4281 5001 5761 81^2
      11 571 25, 29^2 1361 1921 - - 3841 4561 5321 6121 6961 7841
      12 691 27, 881 1441 2041 - - 081 4841 5641 6481 7361 91^2
      13 821 29, 921 39^2 2161 - - - 5121 5961 6841 7761 8721 9721
      14 961 31, 31^2 1601 2281 - - - 5401 6281 7201 8161 9161 101^2
      15 1111 33,1001 41^2 49^2 - - - 5681 6601 7561 8561 9601 10681 11801

      Conjectures:

      21) If any value m1 - 1 found by the processes decribed above is
      substituted for x > 9 in A(x) = 5x^2 + 5X -89 then A(x) is always
      composite.

      Examples:
      If S1 - 9 = 155 and m1 -1 = 28 then A(28) = 3971 = 11*19*19.
      If S1 - 9 = 246 and m1 -1 = 80 then A(80) = 32311 = 79*409.
      If S2 - 9 = 147 and m1 -1 = 47 then A(47) = 11191 =19*19*31.
      If S2 - 9 = 189 and m1 - 1 = 16 then A(16) =1271 = 31*41.

      22) If all possible composites from 21) above are removed from the
      sequence 5x^2 + 5X -89, then all of the remaining values of A(x),
      x > 9, are prime.

      Aldrich Stevens


      C=1
      5*x^4 - 10*x^3 + 10*x^2 - 5*x + 1
      5*x^4 - 10*x^3 + 50*x^2 - 45*x + 11
      5*x^4 - 10*x^3 + 770*x^2 - 765*x + 191
      5*x^4 - 10*x^3 + 13690*x^2 - 13685*x + 3421
      5*x^4 - 10*x^3 + 245530*x^2 - 245525*x + 61381
      5*x^4 - 10*x^3 + 4405730*x^2 - 4405725*x + 1101431

      C=11
      5*x^4 - 10*x^3 + 9548020*x^2 - 9548015*x + 2387011
      5*x^4 - 10*x^3 + 532100*x^2 - 532095*x + 133031
      5*x^4 - 10*x^3 + 29660*x^2 - 29655*x + 7421
      5*x^4 - 10*x^3 + 1660*x^2 - 1655*x + 421
      5*x^4 - 10*x^3 + 100*x^2 - 95*x + 31
      5*x^4 - 10*x^3 + 20*x^2 - 15*x + 11
      5*x^4 - 10*x^3 + 140*x^2 - 135*x + 41
      5*x^4 - 10*x^3 + 2380*x^2 - 2375*x + 601
      5*x^4 - 10*x^3 + 42580*x^2 - 42575*x + 10651
      5*x^4 - 10*x^3 + 763940*x^2 - 763935*x + 190991
      5*x^4 - 10*x^3 + 13708220*x^2 - 13708215*x + 3427061

      C=31
      5*x^4 - 10*x^3 + 16654490*x^2 - 16654485*x + 4163681
      5*x^4 - 10*x^3 + 928130*x^2 - 928125*x + 232091
      5*x^4 - 10*x^3 + 51730*x^2 - 51725*x + 12991
      5*x^4 - 10*x^3 + 2890*x^2 - 2885*x + 781
      5*x^4 - 10*x^3 + 170*x^2 - 165*x + 101
      5*x^4 - 10*x^3 + 50*x^2 - 45*x + 71
      5*x^4 - 10*x^3 + 610*x^2 - 605*x + 211
      5*x^4 - 10*x^3 + 10810*x^2 - 10805*x + 2761
      5*x^4 - 10*x^3 + 193850*x^2 - 193845*x + 48521
      5*x^4 - 10*x^3 + 3478370*x^2 - 3478365*x + 869651
      5*x^4 - 10*x^3 + 62416690*x^2 - 62416685*x + 15604231

      C=41
      5*x^4 - 10*x^3 + 18618670*x^2 - 18618665*x + 4654771
      5*x^4 - 10*x^3 + 1037590*x^2 - 1037585*x + 259501
      5*x^4 - 10*x^3 + 57830*x^2 - 57825*x + 14561
      5*x^4 - 10*x^3 + 3230*x^2 - 3225*x + 911
      5*x^4 - 10*x^3 + 190*x^2 - 185*x + 151
      5*x^4 - 10*x^3 + 70*x^2 - 65*x + 121
      5*x^4 - 10*x^3 + 950*x^2 - 945*x + 341
      5*x^4 - 10*x^3 + 16910*x^2 - 16905*x + 4331
      5*x^4 - 10*x^3 + 303310*x^2 - 303305*x + 75931
      5*x^4 - 10*x^3 + 5442550*x^2 - 5442545*x + 1360741
      5*x^4 - 10*x^3 + 97662470*x^2 - 97662465*x + 24415721

      C=61
      5*x^4 - 10*x^3 + 44561960*x^2 - 44561955*x + 11140721
      5*x^4 - 10*x^3 + 2483360*x^2 - 2483355*x + 621071
      5*x^4 - 10*x^3 + 138400*x^2 - 138395*x + 34831
      5*x^4 - 10*x^3 + 7720*x^2 - 7715*x + 2161
      5*x^4 - 10*x^3 + 440*x^2 - 435*x + 341
      5*x^4 - 10*x^3 + 80*x^2 - 75*x + 251
      5*x^4 - 10*x^3 + 880*x^2 - 875*x + 451
      5*x^4 - 10*x^3 + 15640*x^2 - 15635*x + 4141
      5*x^4 - 10*x^3 + 280520*x^2 - 280515*x + 70361
      5*x^4 - 10*x^3 + 5033600*x^2 - 5033595*x + 1258631
      5*x^4 - 10*x^3 + 90324160*x^2 - 90324155*x + 22581271

      C=71
      5*x^4 - 10*x^3 + 35027630*x^2 - 35027625*x + 8757221
      5*x^4 - 10*x^3 + 1952030*x^2 - 1952025*x + 488321
      5*x^4 - 10*x^3 + 108790*x^2 - 108785*x + 27511
      5*x^4 - 10*x^3 + 6070*x^2 - 6065*x + 1831
      5*x^4 - 10*x^3 + 350*x^2 - 345*x + 401
      5*x^4 - 10*x^3 + 110*x^2 - 105*x + 341
      5*x^4 - 10*x^3 + 1510*x^2 - 1505*x + 691
      5*x^4 - 10*x^3 + 26950*x^2 - 26945*x + 7051
      5*x^4 - 10*x^3 + 483470*x^2 - 483465*x + 121181
      5*x^4 - 10*x^3 + 8675390*x^2 - 8675385*x + 2169161
      5*x^4 - 10*x^3 + 155673430*x^2 - 155673425*x + 38918671
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