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Many 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
    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
    Message 1 of 3 , 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
    • Jack Brennen
      ... It is known that for an odd positive integer A, the following two statements are equivalent: - A is a sum of two squares. - In the prime factorization of
      Message 2 of 3 , Mar 2, 2009
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        Kermit Rose wrote:
        >
        > Which odd positive integers are the sum of two squares?
        >
        > We can't say much about this.
        >

        It is known that for an odd positive integer A, the
        following two statements are equivalent:

        - A is a sum of two squares.

        - In the prime factorization of A, no prime of the form
        4x+3 appears an odd number of times.
      • Kermit Rose
        ... Hello Jack. I overlooked that multiplying a sum of squares times a square also yields a sum of squares. Thus I wish to amend my Function that maps the
        Message 3 of 3 , Mar 2, 2009
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          Jack Brennen wrote:
          >
          > Kermit Rose wrote:
          >>
          >> Which odd positive integers are the sum of two squares?
          >>
          >> We can't say much about this.
          >>
          >
          > It is known that for an odd positive integer A, the
          > following two statements are equivalent:
          >
          > - A is a sum of two squares.
          >
          > - In the prime factorization of A, no prime of the form
          > 4x+3 appears an odd number of times.
          >
          >
          >

          Hello Jack.

          I overlooked that multiplying a sum of squares times a square also
          yields a sum of squares.

          Thus I wish to amend my Function that maps the positive odd sums of two
          squares onto the odd positive integers.

          I had not yet defined to what the function would map squares of primes
          equal to 3 mod 4.

          I extend the function to cover this case by

          F(9) = 3
          F(49) = 7

          F( q**2) = q if q is a prime equal to 3 mod 4.

          Kermit Rose
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