Gaussian primes are Gaussian integers 
satisfying one of the following properties.
1. If both 
and 
are nonzero then, 
is a Gaussian prime iff 
is an ordinary prime.
2. If 
,
then 
is a Gaussian prime iff 
is an ordinary prime and
.
3. If 
,
then 
is a Gaussian prime iff 
is an ordinary prime and
.
The above plot of the complex plane shows the Gaussian
primes as filled squares.
The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (OEIS A002145). The Gaussian primes with 
are given by 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 3, 
, 
, 
, 
, 
, 
, 
, 
, 
.
The numbers of Gaussian primes 
with complex modulus 
(where the definition 
has been used) for
, 1, ... are 0, 100, 4928, 313752,
... (OEIS A091134).
The cover of Bressoud and Wagon (2000) shows an illustration of the distribution
of Gaussian primes in the complex plane.
As of 2009, the largest known Gaussian prime, found in Sep. 2006, is 
, whose real and imaginary parts both have
decimal digits and whose squared
complex modulus has 
digits.