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19874Many to one function which maps positive odd integers which are the sum of two squares onto the positive odd integers

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  • Kermit Rose
    Mar 2, 2009
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      A positive odd prime integer which is equal to 1 mod 4,

      is uniquely represented as the sum of two squares.

      Let p be the name of a prime equal to 1 mod 4.

      Let r and s be the names of the unique positive integers such that
      p = r**2 + s**2.

      Define F(p) = r + s.

      Extend this mapping from the set of primes equal to 1 mod 4, to
      all products of primes equal to 1 mod 4 by


      F(p1**a1 p2**a2 . . . p_k**a_k) = F(p1)**a1 F(p2)**a2 F(p3)**a3 .
      . . F(p_k)**a_k



      F(5) = 2 + 1 = 3
      F(13) = 2 + 3 = 5
      F(17) = 4 + 1 = 5
      F(25) = F(5*5) = F(5)*F(5) = 3 * 3 = 9
      etc

      Which odd positive integers are the sum of two squares?

      We can't say much about this.

      One of the things that we can say is that

      If z is an odd positive integer which is the sum of two squares,

      then one of the squares is odd, and one is even.

      z = ( n-m)**2 + (1 + n + m)**2 = n**2 - 2 n * m + m**2 + 1 + n**2 +
      m**2 + 2 * n + 2 * m + 2 * n * m

      z = 2* n**2 +2 * m**2 + 2 * n + 2 * m + 1

      z = 1 + 4 * sum of two distinct triangular numbers.

      Where one of the triangular numbers is permitted to be equal to zero.

      Does anyone know of (or able to discover) any research previously
      done on this function which maps sums of two opposite parity squares
      onto the odd positive integers ?


      Kermit Rose
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