Loading ...
Sorry, an error occurred while loading the content.

Characterization of positive integers which are the sum of squares in exactly one way.

Expand Messages
  • Kermit Rose
    1d. Re: Formula Posted by: marku606 mark.underwood@sympatico.ca marku606 Date: Sat Oct 3, 2009 9:48 am ((PDT)) ... An integer z is a sum of two squares in
    Message 1 of 1 , Oct 4, 2009
    • 0 Attachment
      1d. Re: Formula
      Posted by: "marku606" mark.underwood@... marku606
      Date: Sat Oct 3, 2009 9:48 am ((PDT))



      --- In primenumbers@yahoogroups.com, "maximilian_hasler" <maximilian.hasler@...> wrote:

      > >
      > >
      >
      >>> > > > *only* primes and powers of primes can be
      >>> > > > expressed as the sum of two squares in only one way.
      >>>
      >> > >
      >> > > 45 = 32 + 62 is not a prime or a power of a prime
      >>
      > >
      > > nor is 10=12+32.
      > > See
      > > http://www.research.att.com/~njas/sequences/A025284
      > > Numbers that are the sum of 2 nonzero squares in exactly 1 way.

      An integer z is a sum of two squares in exactly one way if and only if
      it is

      one of the following forms:

      (1) an odd power of 2, z = 2**(2 * m + 1) for m > 0

      (2) a prime equal to 1 mod 4

      (3) the square of a prime equal to 1 mod 4.

      or

      (4)

      z = p * h**2 where p is either 2, or an odd prime equal to 1 mod 4,
      or the square of an odd prime equal to 1 mod 4,
      and h has all prime factors equal to 3 mod 4.


      The proof follows immediately from the formula

      (a1 **2 + b1**2) * (a2**2 + b2**2) = (a1 a2 - b1 b2)**2 + (a1 b2 + a2
      b1)**2 = (a1 a2 + b1 b2)**2 + (a1 b2 - a2 b1)**2

      and the theorem that a prime equal to 1 mod 4 is the sum of two squares
      in exactly one way.

      Kermit
    Your message has been successfully submitted and would be delivered to recipients shortly.