- --- In email@example.com, "djbroadhurst" <d.broadhurst@...> wrote:
>Yes, of course. :)
> All this is very simply explained
> by unique factorization in K(sqrt(-1)).
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.
- In firstname.lastname@example.org,
Kermit Rose <kermit@...> wrote:
> An integer z is a sum of two squares in exactly one wayThis is clearly false. Simply consider 20 = 2^2 + 4^2.
> 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.
- I wrote:
> An odd number N is the sum of two coprime squares in preciselyand then "Kermit Rose" mistakenly claimed
> one way if and only if N is a prime congruent to 1 mod 4
> or a power of such a prime.
> David, a Minor correction is needed herewith the irrelevant observations
> 2**2 + 11**2 = 5**3Note that 10^2 and 5^2 are not coprime.
> 10**2 + 5**2 = 5**3