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.

REFERENCES:

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.