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Prize Puzzle $$ concerning A = nx^2 + nxy + y^2

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  • Aldrich
    for n = 2,3,6,8, and 14, and the unique issquare function fr^2 - gA = hS^2 associated with each of the binary quadratic forms defined by n. A,x,y,n,r,s,f,g,h :
    Message 1 of 1 , Aug 28, 2010
      for n = 2,3,6,8, and 14, and the unique issquare
      function fr^2 - gA = hS^2 associated with each of
      the binary quadratic forms defined by n.
      A,x,y,n,r,s,f,g,h : integers;

      It is a gray day in Dog Patch. Basically I am
      positing that if a given value of A on one of
      the above equations occurs only once on that
      equation for all (x,y), then that is in itself
      sufficient proof of primality; that A is either
      prime, a square of a prime, or a prime equal to
      A/gcd(A,y^2). At n = 14 the additional condition
      of removing any extraneous factors of 5 also
      applies.

      If a given value of A occurs more then once on
      one of the above equations then A is composite,
      and two factors of A can easily be calculated
      from each (x,y) pair.The reason this works is
      the close connection of x,y to values of r,s of
      issquare. At n = 5, A = 5x^2 + 5xy + y^2
      and the applicable issquare function is
      5r^2 -4A = s^2. if n= 5 and A = 209 = 11* 19,
      then A exists at (1,12) and (5,3) and issquare
      determines which simple algebraic formulas are
      needed to find the r values 14, 15, 13, and 18
      where s is an integer directly from the x,y pairs.
      These r and s values can in turn be used to
      calculate factors of A (no matter how large).

      The binary quadratic forms at n = 2,3,6,8 and 14
      each has its own issquare function that will apply
      to all of its terms. This function will be similar
      but not identical to the function 5r^2 -4A = s^2
      at n = 5.

      A sum of $100 is offered to the first person
      to correctly identify the issquare functions
      that will apply to the binary quadratic forms
      at n = 2,3,6,8, and 14.

      Aldrich Stevens
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