2011/020) Gaussian Integer

Here i discuss Gaussian integer and property of Gaussian integer

x+iy in complex number is a Gaussian integer if x and y are integers

they are closed under addition subtraction and multiplication

A Gaussian integer a + bi is prime if and only if:

* one of a, b is zero and the other is a prime of the form 4n + 3 or its negative − (4n + 3) (where n \geq 0)
* or both are nonzero and a2 + b2 is prime.


13 is not a Gaussian prime because 13 = 9+4 = (3^2+2^2) = (3+2i)(3-2i)

any prime number of the form 4n+1 can be expressed as sum of 2 squares and cannot be a Gaussian prime.

but a prime of the form 4n+3 cannot be expressed as sum of 2 squares and hence Gaussian prime.

2 is a special prime as it is not odd and 2 = 1^2+1^2 = (1+i)(1-i)