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Strange blossoms:

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  • Aldrich
    Speculations concerning binary quadratic forms and issquare functions: The primemine programs that were posted at this PN website needed no reference to
    Message 1 of 1 , Apr 6, 2011
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      Speculations concerning binary quadratic forms
      and issquare functions:

      The primemine programs that were posted at this
      PN website needed no reference to issquare functions
      to identify which integers occurring as values on
      their binary quadratic equations were prime and which
      composite; primes and squares of primes GCD(A,y) = 1
      appear only once as values of A and all its composites
      GCD(A,y) = 1 appear more than once. Nonetheless
      the primary equations of both programs had an underlying
      unique issquare function bound to them which could if
      desired be applied to all of their composites for the
      purpose of factoring. In general, it appears that the
      existence of such a property in binary quadratics is
      quite unusual. This one-to-one correspondence between
      issquare and each term seems likely to be a necessary
      condition for a binary quadratic form to possess if it
      is to be of any use at all in a primality test/factoring
      method. For A = 5*x*x + 5*x*y + y*y the accompanying
      issquare function is 5r*r -4*A = s*s, and for A = 25*x*x +
      25*x*y + y*y it is 21r*r -5*A = s*s.

      Finding the correct issquare function is not a trivial
      problem to solve for many of the eligible equations.
      The issquare function 5r*r -4*A = s*s mentioned above was
      probably the easiest one to discover, is unusually easy to
      apply, and is bound to what is perhaps the most well-
      ordered binary quadratic of them all. It may be observed
      that predictable patterns of many values of the issquare
      function exist and may be calculated from a simple a
      transformation of the x, y coordinates of A (ex: if x = 7,
      y = 6, A = 491 then r1 = 2*7 + 6, r2 = 3*7 + 6 and thus
      5*20*20 - 4*491 = 36 and 5*27*27 - 4*491 = 1681). If 491
      were composite it would exist in at least one more
      location among the values of A, and from the two sets of
      issquare values found factors could easily be calculated.
      Since it is prime it appears only once and indeed because
      it appearsonly once it is prime. Hard proof for the
      behaviors of issquare functions with their paired binary
      quadratic forms are few and far between; a study of them
      is more akin to the science of horticulture than to
      deductive mathematics. One pattern shades into the next,
      properties appear and disappear,and exceptions proliferate
      in some kind of uneven continuum from order to chaos.
      Tinkering with a few rules, and stretching some assumptions
      grows some interesting patterns on the computer screen.

      In many binary quadratic forms there are one or two
      small factors that are not factors of the y variable
      that make some values of A that occur only once look
      composite, a serious breach of our basic rule of
      primality. Since these 'extraneous' factors are few
      and easily handled, however, and the forms in question
      retain the other essential properties, it is logical to
      tolerate an easing of the rule structure and simply
      classify them as unimportant exceptions. It turns out
      that there are quite a few equations with this property
      that can be considered for inclusion into our previous
      list that was expressed as the general equation A =
      N*x*x + N*x*y + y*y. That list now includes N = 2,3,5,7,
      156,165,168,and 264. In addition another general equation
      with a slightly different rules set also looks possibly
      useful in primality test/factoring methods:
      A = 4*N*(x*x -x + x*y) - 2*N*y + 2*y*y + N for N = 3,4,
      5,7,9,11,15,17,23,35,37, and 57. There is no doubt that
      others exist and will eventually discovered.

      The binary quadratic form of primemine2 (A = 25*x*x + 25*x*y
      + y*y ) is an example of an equation with two ignorable
      'extraneous' factors (3 and 7) distributed randomly among
      the A values. These however turned out to be somewhat helpful
      in ascertaining the issquare function bound to it
      (21r*r -5*A = s*s). The application of this function to
      factoring composites of A is far more complex and problematic
      than was the case in pm1 (**for use as a factoring
      method both pm1 and pm2 would need to be adjusted to record
      all x,y values and also add new procedures) and it is not
      even certain that a way to make it work exists in every
      instance. In pm1 finding two locations for a value of A
      would make the finding of factors automatic . In pm2 one of
      four different patterns may need to be used to calculate the necessary issquare values for factoring depending upon whether
      the x,y values for each location of A are odd-odd,odd-even,
      even-odd, or even-even. A small grid of issquare
      values about the origin (x=1,y =1) illustrates
      the situation:

      Perhaps it may be of interest to examine a detailed example
      of how to calculate issquare values at given x,y locations
      in pm2, and how to use two sets of these numbers to find two
      factors of A:

      The value A = 199711 = 41*4871 is found at (27,206) and
      (75,31). The first set of coordinates is odd-even and so
      calculation of issquare begins at (14,6) on the above chart.
      It is not too difficult to figure out what the right values
      should be if one just follows the linear progressions of
      all even rows and all odd columns to the target.
      R1 = 14 + (24 -14)*(27 -1)/2 + (23-14)*(206-2)/2 = 1062,
      S1 * S1 = 21*R1*R1 - 5*A,
      S1 = 4763;
      R2 = 6 + (16 -6)*(27 -1)/2 + (7-6)*(206-2)/2 = 238;
      S2 = 437;
      The second set of coordinates is odd-odd and so
      calculation of issquare begins at (44,8) on the above chart.
      R3 = 44 + (89-44)*(75 -1)/2 + (87-44)*(31-1)/2 = 2354,
      S3 = 1074 ;
      R4 = 8+ (23-8)*(75 -1)/2 + (9-8)*(31-1)/2 = 578
      S4 = 2453;
      Next,three of the S,R pairs are needed to attempt
      the factorization, first calculating:
      (R4 +R2)/(S4 +S2) = 816/2890 = 24/85;
      also calculating (R4 -R2) = 340 and (S4 -S2) = 1996
      and then checking to see if either of these last values
      is divisible by either the numerator or denominator
      from the first calculation. This divisibility requirement
      appears to be a necessary but not sufficient condition
      for a solution to exist, and since 340 mod 85 = 0,
      we can try GCD(A, 816*R1 +/- 340*(24/85)*S1) to see if
      it will yield anything. One factor (41) is found in this
      attempt. If none had been found perhaps a different mix
      of the S,R pairs would yield something. It is unclear if
      a solution always has to exist or not.

      In any event the simple reliable factoring method for all
      composites of A = 5*x*x + 5*x*y + y*y, is replaced with
      a convoluted variant for A = 25*x*x + 25*x*y + y*y that
      must include a proportion other than 1:1 to work , and
      even with that may still not yield any factors. Here the
      proportion is 24/85, but a wide variety is to be expected,
      changing for each new composite being examined.

      Aldrich Stevens
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