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Conjectures Concerning A = 5x^2 + 5xy + y^2

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  • aldrich617
    x, y, A, B : integers; Choose a pair of integers x,y 0 to calculate an A = 5x^2 + 5xy + y^2. Remove all of the prime factors of y that appear in A from A,
    Message 1 of 1 , Aug 13, 2010
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      x, y, A, B : integers;

      Choose a pair of integers x,y > 0 to calculate
      an A = 5x^2 + 5xy + y^2. Remove all of the prime
      factors of y that appear in A from A, with the
      exception primes that end in one or nine, and
      call this value B. If the A value appears two
      or more times on 5x^2 + 5xy + y^2, then B is
      composite. (*see note). If A appears only once
      then B is prime or a perfect square.

      Examples:

      If x = 1 and y = 11 then A = 181 and B = 181.
      The value A does not occur again, so B must be
      a prime or perfect square.

      If x = 2 and y = 10 then A = 220 and B = 11.
      The value A does not occur again, so B must be
      a prime or perfect square.

      If x = 12 and y = 16 then A = 1936 and B = 121.
      The value A does not occur again, so B must be
      a prime or perfect square.

      If x = 1 and y = 12 then A = 209 and B = 209.
      The value A does occur again at x = 5 and y = 3,
      so B must be a composite.

      If x = 22 and y = 11 then A = 3751 and B = 3751
      (11*11*31). The value A does occur again at x = 5
      and y = 49 and also at x = 15 and y = 26,
      so B must be a composite.

      If x = 1 and y = 605 then A = 369055 and B = 73811.
      The value A does occur again at x = 99
      and y = 370, so B must be a composite.


      *Note: two factors of a given A can easily be calculated
      if two sets of x,y pairs exist for it.

      Aldich Stevens
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