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From: quasi on 9 Aug 2010 04:49
Can a prime be a sum of 2 squares in more than one way?

quasi
From: achille on 9 Aug 2010 05:22
On Aug 9, 4:49 pm, quasi <qu...(a)null.set> wrote:
> Can a prime be a sum of 2 squares in more than one way?
>
> quasi

Are you taking about rational primes? then the answer is NO.

This is because the gaussian integers Z[i] is an UFD.
If a rational prime p can be written as a sum of two squares:

p = a^2 + b^2 = (a + bi)(a - bi) a, b \in Z\{0},

then it is easy to see a+bi and a-bi irreducible and hence
are primes in Z[i], the "unique factorization" property of
Z[i] then rule out other decomposition of p essentially
differ from this.






From: quasi on 9 Aug 2010 11:43
On Mon, 9 Aug 2010 02:22:38 -0700 (PDT), achille
<achille_hui(a)yahoo.com.hk> wrote:

>On Aug 9, 4:49 pm, quasi <qu...(a)null.set> wrote:
>> Can a prime be a sum of 2 squares in more than one way?
>>
>> quasi
>
>Are you taking about rational primes? then the answer is NO.
>
>This is because the gaussian integers Z[i] is an UFD.
>If a rational prime p can be written as a sum of two squares:
>
> p = a^2 + b^2 = (a + bi)(a - bi) a, b \in Z\{0},
>
>then it is easy to see a+bi and a-bi irreducible and hence
>are primes in Z[i], the "unique factorization" property of
>Z[i] then rule out other decomposition of p essentially
>differ from this.

Thanks.

I missed the natural tie to Z[i].

quasi
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