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The answer key for the Prize Puzzle $$ of 8/29/10.

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  • Aldrich
    It concerns A = nx^2 + nxy + y^2 for n = 2,3,5,6,8, and 14, and the unique issquare function fr^2 - gA = hS^2 associated with each of the binary quadratic
    Message 1 of 1 , Sep 14, 2010
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      It concerns A = nx^2 + nxy + y^2
      for n = 2,3,5,6,8, 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,x1,y1,x2,y2,
      r1,r2,r3,r4,s1,s2,s3,s4,j1,j2 : integers;

      At n = 2, A = 2x^2 + 2xy + y^2 and its
      applicable issquare function is -r^2 + 2A = s^2.

      At n = 3, A = 3x^2 + 3xy + y^2 and its
      applicable issquare function is -3r^2 + 4A = s^2.

      At n = 5, A = 5x^2 + 5xy + y^2 and its
      applicable issquare function is 5r^2 - 4A = s^2.

      At n = 6, A = 6x^2 + 6xy + y^2 and its
      applicable issquare function is 3r^2 - 2A = s^2.

      At n = 8, A = 8x^2 + 8xy + y^2 and its
      applicable issquare function is 2r^2 -A = s^2.

      At n = 14, A = 14x^2 + 14xy + y^2 and its
      applicable issquare function is 7r^2 - 2A = 5s^2.

      Example at n = 14:
      Two values each for r and s, namely r1,r2,s1,s2, may
      be obtained directly from the x,y coordinates (x1,y1)
      of any A by means of simple formulas:
      r1 = 2x1 + y1,
      r2 = r1 + 10x1,
      s1 = y1,
      s2 = r1 + r2 -s1
      If an identical A value exists with different x,y
      coordinates, then values for r3,r4,s3,and s4 may
      found in the same way, and two factors j1,j2 of A
      may be easily calculated:
      j1,j2 = GCD(A, (r1 + r2)*r3 +/- (r2 - r1)*s3).
      If A = 40889 then x1 =32, y1 =53, x2 =37, y2 =39
      and the two factors are 31 and 1319.

      Where have all the prize hawks gone,long time passing?
      Doesn't anyone believe in a free lunch anymore?
      It is another gray day in Dog Patch.

      Aldrich Stevens
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