EDIT: When I wrote this post originally, I was not careful about the distinction between

and

. Here's what I wrote, slightly edited:

~~~~~~~~~~~~~~~~~~~~~~

tonio's first post (post #2 of this thread) was not "generic" but was a direct and concrete answer to your questions. To spell it out more explicitly,

(1) The answer is, there does not exist any such Gaussian prime.

Further reading:

Gaussian Prime -- from Wolfram MathWorld Gaussian integer - Wikipedia, the free encyclopedia (subheading: As a unique factorization domain)

Note: I don't know why tonio mentions

separately, as it fits the description

where

with

a regular prime in

.

Correct. I was just thinking of primes which are 1 modulo 4 when I wrote , and tried to distinguish the first prime 2 which isn't 1 modulo 4 and still fulfills the condition: , but I didn't write down the distinction.

All the above was thought to convey the fact (theorem) that a prime is a sum of squares iff it is 2 or it is 1 mod 4 , and from here to deduce, together with a little algebra, that an integer is a sum of squares iff any prime equal to 3 mod 4 that appears in its prime decompostion appears there to an even power.

Thanx

Tonio
(2) tonio gave you the prime factorisation of 6 in

. Note that 2*3 is not a prime factorisation here because 2 is not a Gaussian prime.

Further reading:

Unique factorization domain - Wikipedia, the free encyclopedia Unique Factorization Domain -- from Wolfram MathWorld
~~~~~~~~~~~~~~~~~~~~~~

So, the above applies to

. Question to the OP (Samson): Are you sure you want to know about primes in

as opposed to

? If so, I'll have to think/research more, or someone else will have to add some info..