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Gaussian Prime

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GaussianPrimes

A Gaussian prime is a Gaussian integer z=a+bi satisfying one of the following properties.

1. If both a and b are nonzero, then a+bi is a Gaussian prime iff a^2+b^2 is an ordinary prime.

2. If a=0, then bi is a Gaussian prime iff |b| is an ordinary prime and |b|=3 (mod 4).

3. If b=0, then a is a Gaussian prime iff |a| is an ordinary prime and |a|=3 (mod 4).

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 |a|,|b|<=5 are given by -5-4i, -5-2i, -5+2i, -5+4i, -4-5i, -4-i, -4+i, -4+5i, -3-2i, -3, -3+2i, -2-5i, -2-3i, -2-i, -2+i, -2+3i, -2+5i, -1-4i, -1-2i, -1-i, -1+i, -1+2i, -1+4i, -3i, 3i, 1-4i, 1-2i, 1-i, 1+i, 1+2i, 1+4i, 2-5i, 2-3i, 2-i, 2+i, 2+3i, 2+5i, 3-2i, 3, 3+2i, 4-5i, 4-i, 4+i, 4+5i, 5-4i, 5-2i, 5+2i, 5+4i.

The numbers of Gaussian primes z with complex modulus |z|<=10^n (where the definition |a+ib|=sqrt(a^2+b^2) has been used) for n=0, 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.

Large Gaussian primes having nonzero real part and imaginary part include the Gaussian Mersenne primes (1+i)^n-1 (Caldwell).

The Gaussian moat problem has a negative answer. No infinite sequence of distinct Gaussian primes has bounded consecutive Euclidean distances (OpenAI 2026).

See also

Eisenstein Prime, Gaussian Integer, Gaussian Mersenne Prime, Gaussian Moat Problem, Moat-Crossing Problem, Prime Number

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References

Bressoud, D. M. and Wagon, S. A Course in Computational Number Theory. London, England: Springer-Verlag, 2000.Caldwell, C. "Gaussian Mersenne Norm." https://t5k.org/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 k(i)" and "The Fundamental Theorem of Arithmetic in k(i)." §12.7 and 12.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 183-187, 1979.OpenAI. "Bounded-Step Walks on Gaussian Primes." 26 Sep 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf.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." https://www.oocities.org/hjsmithh/GPrimes/index.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, England: Penguin, p. 85, 1991.Zariski, O. and Samuel, P. Commutative Algebra I. New York: Springer-Verlag, 1958.

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Gaussian Prime

Cite this as:

Weisstein, Eric W. "Gaussian Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GaussianPrime.html

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