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Re: Formula

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  • marku606
    ... Yes, of course. :) Simplicity is one thing; understanding the simple can be downright elusive to the untrained. Thank you David for your explanation. By
    Message 1 of 11 , Oct 3 6:44 PM
      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
      >

      > All this is very simply explained
      > by unique factorization in K(sqrt(-1)).

      Yes, of course. :)

      Simplicity is one thing; understanding the simple can be downright elusive to the untrained. Thank you David for your explanation. By the powers of heaven, one day I'm going to understand it.

      Mark
    • djbroadhurst
      In primenumbers@yahoogroups.com, ... This is clearly false. Simply consider 20 = 2^2 + 4^2. David
      Message 2 of 11 , Oct 4 10:01 AM
        In primenumbers@yahoogroups.com,
        Kermit Rose <kermit@...> wrote:

        > 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.

        This is clearly false. Simply consider 20 = 2^2 + 4^2.

        David
      • djbroadhurst
        ... and then Kermit Rose mistakenly claimed ... with the irrelevant observations ... Note that 10^2 and 5^2 are not coprime. David
        Message 3 of 11 , Oct 4 10:13 AM
          I wrote:

          > An odd number N is the sum of two coprime squares in precisely
          > one way if and only if N is a prime congruent to 1 mod 4
          > or a power of such a prime.

          and then "Kermit Rose" mistakenly claimed

          > David, a Minor correction is needed here

          with the irrelevant observations

          > 2**2 + 11**2 = 5**3
          > 10**2 + 5**2 = 5**3

          Note that 10^2 and 5^2 are not coprime.

          David
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