Gaussian Prime
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.
Bressoud, D. M. and Wagon, S. A Course in Computational Number Theory. London: Springer-Verlag, 2000.
Caldwell, C. "Gaussian Mersenne Norm." http://primes.utm.edu/top20/page.php?id=41.
Gethner, E.; Wagon, S.; and Wick, B. "A Stroll Through the Gaussian Primes." Amer. Math. Monthly 105, 327-337, 1998.
Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes." §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33-36, 1994.
Hardy, G. H. and Wright, E. M. "Primes in 
" and "The
Fundamental Theorem of Arithmetic in 
." §12.7
and 12.8 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 183-187, 1979.
Rademacher, H. Topics in Analytic Number Theory. New York: Springer-Verlag, 1973.
Sloane, N. J. A. Sequences A002145/M2624, A091100, and A091134 in "The On-Line Encyclopedia of Integer Sequences."
Smith, H. J. "Gaussian Primes." http://www.geocities.com/hjsmithh/GPrimes.html.
Wagon, S. "Gaussian Primes." §9.4 in Mathematica in Action. New York: W. H. Freeman, pp. 298-303, 1991.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 85, 1991.
Zariski, O. and Samuel, P. Commutative Algebra I. New York: Springer-Verlag, 1958.
Weisstein, Eric W. "Gaussian Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GaussianPrime.html
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